Device and method for pressure control of electric injection molding machine

ABSTRACT

The exact method with small time-lag of detecting injection pressure for controlling pressure in an electric-motor driven injection molding machine without using a pressure detector has been asked for because the pressure detector is very expensive, necessitates troublesome works for mounting, an electric protection against noise and the works for zero-point and span adjustings and causes a complicate mechanical structure. The present invention uses a high-gain observer which contains the discrete-time arithmetic expressions derived from a mathematical model of an injection and pressure application mechanism in an electric-motor driven injection molding machine consisting of state equations and outputs an estimate of injection pressure, which is one of the state variables of the above state equations, by using a screw position signal and a servomotor current demand signal or actual motor current signal as inputs. The high-gain observer obtains the exact injection pressure estimate with very small time-lag without using a pressure detector. Thus the estimate of injection pressure fed by the high-gain observer can be adopted as a feedback signal of actual injection pressure for controlling injection pressure.

TECHNICAL FIELD

This invention is concerning an apparatus and a method for controllingpressure in an electric-motor driven injection molding machine.

BACKGROUND ART

AC servomotors are becoming used for middle-sized injection moldingmachines heretofore driven by hydraulic actuators (clamping force>3.5MN) that have high precision, quick response and higher power which areobtained by performance improvements of permanent magnets and costreductions.

An injection molding machine consists of a plasticizier in which resinpellets are melted by friction heat generated by plasticizing screwrevolution and stored at the end of a barrel, an injector in which anamount of melted polymer is injected into a metal mold at a givenvelocity and a given dwell pressure is applied, and a clamper in whichthe metal mold is clamped and opened, all using AC servomotors drivesystem. FIG. 3 shows a view of an injection molding mechanism using anAC servomotor.

On an injection machine base which is fixed on the ground, a movablebase is located which moves on a linear slider and both the bases arenot shown in FIG. 3. All parts except a metal mold 1 shown in FIG. 3 aremounted on the movable base. By sliding the movable base, the top of abarrel 2 is clamped on the metal mold 1 and vice versa the top of thebarrel 2 is separated from the metal mold 1. FIG. 3 shows a mode inwhich the top of the barrel 2 is clamped on the metal mold 1 beforemelted polymer being injected into the metal mold 1.

On the movable base, a servomotor 3, a reduction gear 4, a ball screw 5and a bearing 6 are fixed. A nut 7 of the ball screw 5, a moving part 8,a screw 9 and a pressure detector 10 such as a load cell consist of anintegral structure. The moving part 8 is mounted on a linear slider 11so that the integral structure is moved back and forth by the movementof the nut 7.

Rotation of the servomotor 3 is transferred to the ball screw 5 whichmagnifies linear force through the reduction gear 4 and rotation of theball screw 5 is converted to a linear motion of the nut 7 of the ballscrew 5 and a linear motion of the screw 9 and pressure application tomelted polymer are realized through the moving part 8. Position of thescrew 9 is detected by a rotary encoder 12 mounted on the servomotor 3.Pressure applied to the melted polymer at the end of the barrel 2 isdetected by the pressure detector 10 mounted between the nut 7 and themoving part 8. A cavity 13 in the metal mold 1 is filled up with meltedpolymer by a movement of the screw 9.

Mold good manufacturing consists of injection and dwell pressureapplication. In the injection process, polymer melt must be injectedinto the cavity 13 as fast as possible so that temperatures of polymerin the cavity become homogeneous. However, as excessive injectionvelocity brings about excessive polymer pressure and mold defects,polymer pressure in the injection process is constrained under a givenpressure limit pattern. In the pressure application process followingthe injection process, a given pressure pattern is applied for eachgiven duration at the polymer in the cavity during cooling in order tosupply a deficiency due to polymer shrinkage. Therefore, the followingtwo requirements are given to the injection velocity pattern and thepressure application pattern.

-   -   (1) In the injection process, a given injection velocity pattern        is realized and at the same time injection pressure is        constrained under a given pressure limit pattern in terms of        mold good quality.    -   (2) In the pressure application process, a given pressure        pattern is realized and at the same time injection velocity is        constrained under a given velocity limit pattern in terms of        safety operation.

In the injection process (time 0˜t₁) shown in FIG. 4( a), injectionvelocity control is carried out by giving injection velocity commandshown in FIG. 4( b) to realize a given injection velocity pattern.However, injection pressure has to be controlled lower than a givenpressure limit pattern shown in FIG. 4( c). Vertical scales 100% shownin FIG. 4( b) and (c) indicate maximum values of injection velocity andinjection pressure, respectively.

In the pressure application process (time t₁˜t₂) shown in FIG. 4( a),pressure application control is carried out by giving pressureapplication command shown in FIG. 4( c) to realize a given pressureapplication pattern. However, injection velocity has to be controlledlower than a given injection velocity limit pattern shown in FIG. 4( b).

FIG. 5 shows a block diagram of a controller which realizes the abovetwo requirements (1) and (2) (paragraph{0008}) (patent literature PTL1). The controller consists of an injection controller 20 and a motorcontroller (servoamplifier) 40.

The injection controller 20 executes a control algorithm at a constanttime interval Δt and a discrete-time control is used. The injectioncontroller 20 consists of an injection velocity setting device 21, atransducer 22, a pulse generator 23, an analog/digital (A/D) converter25, an injection pressure setting device 26, a subtracter 27, a pressurecontroller 28, and a digital/analog (D/A) converter 29. The pressuredetector 10 is connected to the injection controller 20.

The injection velocity setting device 21 feeds a time sequence ofinjection velocity command V_(i)* to the transducer 22. The transducer22 calculates screw displacement command Δx_(v)* for the screw 9 whichhas to move during the time interval Δt by the following equation (1).

{Math. 1}

Δx_(v)* =V_(i)*Δt   (1)

The command Δx_(v)* is fed to the pulse generator 23.

The pulse generator 23 feeds a pulse train 24 corresponding to thecommand Δx_(v)* . The pulse train 24 is fed to a pulse counter 41 in themotor controller 40.

The pressure detector 10 feeds an injection pressure signal P_(i) to theinjection controller 20 through the A/D converter 25. The A/D converter25 feeds the pressure signal P_(i) to the subtracter 27.

The injection pressure setting device 26 feeds a time sequence ofinjection pressure command P_(i)* to the subtracter 27. The subtracter27 calculates a pressure control deviation ΔP_(i) by the followingequation (2).

{Math. 2}

ΔP _(i) =P _(i)*−P_(i)   (2)

The control deviation ΔP_(i) is fed to the pressure controller 28.

The pressure controller 28 calculates a motor current demand i_(p)* fromΔP_(i) by using PID (Proportional+Integral+Derivative) control algorithmand feeds the demand i_(p)* to the motor controller 40 through the D/Aconverter 29.

The motor controller 40 consists of pulse counters 41 and 44, an A/Dconverter 42, a comparator 43, subtracters 45 and 48, a positioncontroller 46, a differentiator 47, a velocity controller 49 and a PWM(Pulse Width Modulation) device 50. The motor controller 40 is connectedto the servomotor 3 equipped with the rotary encoder 12. The demandi_(p)* is fed to the comparator 43 through the A/D converter 42.

The pulse counter 41 accumulates the pulse train 24 from the injectioncontroller 20 and obtains screw position demand x* and feeds the demandx* to the subtracter 45. The pulse counter 44 accumulates the pulsetrain from the rotary encoder 12 and obtains actual screw position x andfeeds the position signal x to the subtracter 45.

The subtracter 45 calculates a position control deviation (x*−x) andfeeds the position deviation to the position controller 46. The positioncontroller 46 calculates velocity demand v* by the following equation(3) and feeds the demand v* to the subtracter 48.

{Math. 3}

v*=K _(p)(x*−x)   (3)

where K_(p) is a proportional gain of the position controller 46.

The rotary encoder 12 feeds a pulse train to the differentiator 47 andto the pulse counter 44. The differentiator 47 detects actual screwvelocity v and feeds the velocity signal v to the subtracter 48.

The subtracter 48 calculates a velocity control deviation (v*−v) andfeeds the velocity deviation to the velocity controller 49. The velocitycontroller 49 calculates a motor current demand i_(v)* by the followingequation (4) and feeds the demand i_(v)* to the comparator 43.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 4} \right\} & \; \\{i_{v}^{*} = {{K_{Pv}\left( {v^{*} - v} \right)} + {\frac{K_{Pv}}{T_{Iv}}{\int{\left( {v^{*} - v} \right){t}}}}}} & (4)\end{matrix}$

where K_(Pv) and T_(Iv) are a proportional gain and an integral timeconstant of the velocity controller 49, respectively. In the motorcontroller 40 a position control loop has a minor loop of velocitycontrol.

The comparator 43 to which both motor current demands i_(v)* and i_(p)*from the velocity controller 49 and the pressure controller 28,respectively, are fed, selects a lower current demand i* of i_(v)* andi_(p)* and feeds the lower demand i* to the PWM device 50. The PWMdevice 50 applies three-phase voltage to the servomotor 3 so that theservomotor 3 is driven by the motor current i*. The comparator 43restricts motor current demand i_(v)* decided by injection velocitycontrol loop to motor current demand i_(p)* decided by pressure controlloop.

In FIG. 4, transfer time t₁ from injection to pressure application isspecified by an operator, so the time t₁ should coincide with the timeat which the cavity is filled up with polymer melt, but it is difficultfor an operator to set the time t₁ at the exact time. However, it can beshown that the above two requirements (1) and (2) (paragraph{0008}) arerealized by the comparator 43. Firstly the finishing time t₁ ofinjection process is supposed to be set by an operator before the timeat which the cavity 13 is filled up with polymer melt actually. When thetime reaches t₁ and pressure application process starts, actual pressureP_(i) is lower than a set value P_(i)* because the cavity is not yetfilled and motor current demand i_(p)* increases so that pressure P_(i)is increased to the set value P_(i)*.

If demand i_(p)* is selected, injection velocity increases rapidlybecause the cavity is still filling. Actual velocity could exceed thevelocity limit shown in FIG. 4 (b), failing requirement (2)(paragraph{0008}). Even if demand i_(p)* exceeds i_(v)* when pressureapplication starts, however, the comparator 43 selects the lower demandi_(v)* and limits velocity so requirement (2) is met, that is, by thecomparator 43 pressure application is transferred to velocity limit inorder to satisfy requirement (2).

Secondly the time t₁ is supposed to be set after the time at which thecavity 13 is filled up actually. Even when the cavity is filled up,injection process continues, but actual screw speed slows as the cavityis filled and motor current demand i_(v)* is increased to maintaininjection velocity.

If demand i_(v)* is selected by the comparator 43, injection pressureincreases rapidly because filling is completed and actual pressure mayexceed the pressure limit shown in FIG. 4 (c), failing requirement (1)(paragraph{0008}). Even if demand i_(v)* exceeds i_(p)* in injectionprocess, however, the comparator 43 selects the lower demand i_(p)* andlimits pressure control in injection, so requirement (1) is met, thatis, by the comparator 43 velocity control is transferred to pressurelimit in order to satisfy requirement (1).

In the controller shown in FIG. 5, the pressure detector 10 isabsolutely necessary. Patent literatures PTL 2˜PTL 9 are applications ofthe pressure control apparatus and method of injection molding machineswithout using the pressure detector.

In patent literature PTL 2 for hydraulic actuator driven injectionmachines, polymer charateristics formula which gives the relationalexpression among polymer pressure, polymer temperature and polymerspecific volume is used and the required polymer pressure is calculatedby inputting measured polymer temperature and polymer specific volumewhich is decided from the desired value of mold good weight. Then byusing initial temperatures of metal mold and polymer at the start ofpressure application process and the above required polymer pressure,the required set value of pressure application is derived through anapproximate expression. The pressure application set value is fed to thehydraulic servovalve amplifier as the voltage command converted and theset value of applied pressure is realized by the hydraulic pressure ofhydraulic cylinder piston.

In patent literature PTL 3, in order to detect polymer pressure in thecavity the pressure is applied to a plunger which moves back and forthin the cavity and is connected with a ball screw mechanism whose nut isrotated by a servomotor. In injection and pressure application processthe servomotor holds the position of the plunger to which the polymerpressure is applied and the servomotor current is detected by a currenttransducer and the detected current is converted to the polymer pressurein the cavity. The position of the plunger is detected by a rotaryencoder equipped with the servomotor.

In patent literature PTL 4, in order to detect polymer pressure in thecavity a disturbance observer is used for a servomotor drive systemwhich moves a plunger back and forth in the cavity. In injection andpressure application process the pressure is applied to the plunger andthe servomotor drive system holds the position of the plunger. Then thedisturbance observer estimates load torque of the servomotor by using amotor speed signal and a motor torque command signal. The pressure inthe cavity is obtained from the estimated load torque. The arithmeticexpressions of the observer are shown in the literature. The method bywhich pressure in the cavity is obtained directly by using detectedservomotor current or motor torque command, is also shown in theliterature.

In patent literature PTL 5, firstly a function which estimates polymerpressure in the cavity by using injection screw drive force andinjection velocity, is decided. In the actual control actions, featuresize of mold good, polymer data, real-time data of screw drive force andinjection velocity are fed to the above function and the real-timeestimated pressure in the cavity is obtained. Injection velocity iscontrolled by deviation of the estimated pressure from the referencevalue. The procedures of obtainig the above exact function are shown inthe literature.

In patent literature PTL 6, the pressure control apparatus is realizedin which an observer for a servomotor drive system estimates polymerpressure and the estimated pressure is used as a detected signal for thepressure control. The observer is fed by a motor speed in an injectionprocess and the total friction resistance in an injection mechanism andoutputs the estimates of motor speed and polymer pressure. The observeris applied to the following two models.

-   -   (1) A servomotor drives an injection screw through a linear        motion converter such as a ball screw only.    -   (2) A servomotor drives an injection screw through a belt pulley        reduction gear and a linear motion converter.

In the observer model (1), the friction resistance consists of a dynamicfriction resistance and a static friction resistance over an injectionmechanism. In the observer model (2), the friction resistance consistsof a dynamic friction resistance only which is defined as a sum of avelocity dependent component and a load dependent component.

In the observer model (1) the polymer pressure is assumed to beconstant. In the observer model (2) the observer outputs not only theestimates of motor speed and polymer pressure but also the estimates ofpulley speed at load side, belt tension and force applied to polymermelt by a screw. It is assumed that the belt is elastic and the timerate of change in polymer pressure is proportional to pulley speed atload side, to pulley acceleration and to force applied to polymer by ascrew. The force applied to polymer melt by a screw is assumed to beconstant.

In patent literature PTL 7, an injection velocity and pressure controlapparatus is realized in which an observer for a servomotor drive systemestimates the load torque generated by polymer pressure. The observer isfed by motor speed signal and motor current command signal and outputsthe estimates of motor speed, load torque and position differencebetween a motor shaft and a load side shaft. The model of the observerconsists of the servomotor drive system which moves a screw through abelt pulley reduction gear. The pressure value converted from theestimated load torque obtained by the observer is used as a detectedpressure signal.

In patent literature PTL 8, the observer for the model (2) in PTL 6(paragraph {0041}) is used and the pressure control method is invented.The method uses motor speed and the estimates of pulley speed at loadside, belt tension and polymer pressure as feedback signals of statevariables. These estimates are obtained by the observer. Another controlmethod is invented, in which the servomotor torque command is decided bythe above four state variables.

In patent literature PTL 9, a method is invented, in which the estimateof load torque applied by polymer pressure is obtained by using aninverse model of a transfer function whose inputs are motor generatedtorque and load torque and output is motor speed. The inverse model isfed by motor speed and motor generated torque and derives the estimateof load torque. The polymer pressure is obtained from the estimate ofthe load torque. The inverse model requires the high-orderdifferentiation.

The common object of inventions described in patent literatures PTL2˜PTL 9 which detect polymer pressure without using a pressure detectoris to avoid the following disadvantages.

-   -   (1) A highly reliable pressure detector is very expensive under        high pressure circumstances.    -   (2) Mounting a pressure detector in the cavity or the barrel        nozzle part necessitates the troublesome works and the working        cost becomes considerable.    -   (3) Mounting a load cell in an injection shafting alignment from        a servomotor to a screw complicates the mechanical structure and        degrades the mechanical stiffness of the structure.    -   (4) A load cell which uses strain gauges as a detection device        necessitates an electric protection against noise for weak        analog signals. Moreover the works for zero-point and span        adjustings of a signal amplifier are necessary (patent        literature PTL 10).

Patent literatures PTL 11 and PTL 12 are inventions concerning thepressure control at electric-motor driven injection molding machines andboth necessitate pressure detectors. In patent literature PTL 11, theconcept of virtual screw velocity ω₁ is introduced in equation (1) ofthe description of PTL 11 and equation (1) is based on the point of viewwhich the pressure control is conducted by screw position control. Anexact control method is realized by using virtual velocity ω₁ as aparameter which compensates the pressure loss due to the nonlinear losswhich results in the difference between the pressure corresponding tomotor generated torque and pressure set value. The disturbance observeroutputs the estimate of virtual velocity ω₁ so that the differencebetween the pressure detected by a load cell and the estimated pressurebecomes zero by using the error between detected pressure and theestimated pressure. Patent literature PTL 12 is a prior application ofPTL 11 and it is different from PTL 11 in the observer structure.

CITATION LIST Patent Literature

PTL 1: Patent No.3787627

PTL 2: Patent 5-77298

PTL 3: Patent 6-856

PTL 4: Patent 7-299849

PTL 5: Patent 9-277325

PTL 6: Patent W02005/028181

PTL 7: Patent 2006-142659

PTL 8: Patent 2006-256067

PTL 9: Patent 2008-265052

PTL 10: Patent 2003-211514

PTL 11: Patent 10-244571

PTL 12: Patent 10-44206

Non Patent Literature

NPL 1: H. K. Khalil, Nonlinear Systems, 14.5 High-Gain Observers,Prentice-Hall, (2002), pp. 610-625

NPL 2: B. D. O. Anderson and J. B. Moore, Optimal Control, LinearQuadratic Methods, 7.2 Deterministic Estimator Design, Prentice-Hall,(1990), pp. 168-178

NPL 3: A. M. Dabroom and H. K. Khalil, Discrete-time implementation ofhigh-gain observers for numerical differentiation, Int. J. Control, Vol.72, No. 17, (1999), pp. 1523-1537

NPL 4: A. M. Dabroom and H. K. Khalil, Output Feedback Sampled-DataControl of Nonlinear Systems Using High-Gain Observers, IEEE Trans.Automat. Contr., Vol. 46, No. 11, (2001), pp. 1712-1725

SUMMARY OF INVENTION Technical Problem

The problem that starts being solved is to realize a pressure controlapparatus and a pressure control method of electric-motor driveninjection molding machines which satisfy the above two requirements (1)and (2) (paragraph{0008}) without using a pressure detector in order toavoid the four disadvantages described in Background Art(paragraph{0046}) resulted by using a pressure detector.

SOLUTION TO PROBLEM

Mold good manufacturing consists of injection and dwell pressureapplication. In the injection process injection pressure has to beconstrained under a given pressure limit pattern by the requirement (1)(paragraph{0008}) and so it is necessary to detect the actual pressurewithout time-lag. In the pressure application process a given pressurepattern has to be realized by the requirement (2) (paragraph{0008}) andso it is necessary to detect the applied pressure without time-lag.Therefore a pressure detecting means is required to have no time-lag.

As the error of detected pressure causes mold good defects and lack ofsafety operation, the exact pressure detection is required. Thereforethe method of a high-gain observer (non patent literature NPL 1) is usedto realize a pressure detecting means which satisfies the following tworequirements (A) and (B).

(A) The detection means has very small time-lag.

(B) The detection means is high-precision.

A high-gain observer estimates all state variables by using detectedvariables. This is explained by using a simple mathematical model asfollows. Equation (5) shows a state equation of a simple model.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 5} \right\} & \; \\\left. \begin{matrix}{{\overset{.}{x}}_{1} = x_{2}} \\{{\overset{.}{x}}_{2} = {\varphi \left( {x,u} \right)}} \\{y = x_{1}} \\{x = \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}}\end{matrix} \right\} & (5)\end{matrix}$

where x₁, x₂: State variables, u: Input variable, y: Output variable,φ(x, u): nonlinear function of variables x, u. For example x₁ isposition variable, x₂ is velocity variable and u is motor currentvariable. Output variable y and input variable u are supposed to bemeasurable. The high-gain observer which estimates state x is given byequation (6).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 6} \right\} & \; \\\left. \begin{matrix}{{{\overset{\overset{.}{\hat{}}}{x}}_{1}{\hat{x}}_{2}} + {H_{1}\left( {y - {\hat{x}}_{1}} \right)}} \\{{\overset{\overset{.}{\hat{}}}{x}}_{2} = {{\varphi_{0}\left( {\hat{x},u} \right)} + {H_{2}\left( {y - {\hat{x}}_{1}} \right)}}}\end{matrix} \right\} & (6)\end{matrix}$

where {circumflex over (x)}₁, {circumflex over (x)}₂: Estimates of statevariables x₁, x₂, H₁, H₂: Gain constants of the high-gain observer whichare larger than 1, φ₀: Nominal function of φ used in the high-gainobserver computing. Estimation errors {tilde over (x)}₁, {tilde over(x)}₂ by using the high-gain observer (6) are given by equation (7) fromequations (5) and (6).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 7} \right\} & \; \\\left. \begin{matrix}{{\overset{\overset{.}{\sim}}{x}}_{1} = {{{- H_{1}}{\overset{\sim}{x}}_{1}} + {\overset{\sim}{x}}_{2}}} \\{{\overset{\overset{.}{\sim}}{x}}_{2} - {H_{2}{\overset{\sim}{x}}_{1}} + {\delta \left( {x,\hat{x},u} \right)}}\end{matrix} \right\} & (7) \\\left. \begin{matrix}{{\overset{\sim}{x}}_{1} = {x_{1} - {\hat{x}}_{1}}} \\{{\overset{\sim}{x}}_{2} = {x_{2} - {\hat{x}}_{2}}} \\{{\delta \left( {x,\hat{x},u} \right)} = {{\varphi \left( {x,u} \right)} - {\varphi_{0}\left( {\hat{x},u} \right)}}}\end{matrix} \right\} & (8)\end{matrix}$

where δ: Model error between the nominal model φ₀ and the true butactually unobtainable function φ. Introducing a positive parameter εmuch smaller than 1, H₁, H₂ are given by equation (9).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 8} \right\} & \; \\{H_{1} = {{\frac{K_{1}}{ɛ}\mspace{14mu} H_{2}} = \frac{K_{2}}{ɛ^{2}}}} & (9)\end{matrix}$

As H₁, H₂ in equation (9) are large gain constants, equation (6) iscalled by a high-gain observer. By using equation (9), equation (7) isrewritten as equation (10).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 9} \right\} & \; \\\left. \begin{matrix}{{\overset{\overset{.}{\sim}}{x}}_{1} = {{- {K_{1}\left( {{\overset{\sim}{x}}_{1}/ɛ} \right)}} + {\overset{\sim}{x}}_{2}}} \\{{\overset{\overset{.}{\sim}}{x}}_{2} = {{{- \left( {K_{2}/ɛ} \right)}\left( {{\overset{\sim}{x}}_{1}/ɛ} \right)} + {\delta \left( {x,\overset{\sim}{x},u} \right)}}}\end{matrix} \right\} & (10)\end{matrix}$

The estimation errors {tilde over (x)}₁, {tilde over (x)}₂ are replacedby new variables η₁, η₂ as written in equation (11).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 10} \right\} & \; \\{\eta_{1} = {{\frac{{\overset{\sim}{x}}_{1}}{ɛ}\mspace{14mu} \eta_{2}} = {\overset{\sim}{x}}_{2}}} & (11)\end{matrix}$

By using equation (11), equation (10) is rewritten as equation (12).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 11} \right\} & \; \\\left. \begin{matrix}{{ɛ{\overset{.}{\eta}}_{1}} = {{{- K_{1}}\eta_{1}} + \eta_{2}}} \\{{ɛ{\overset{.}{\eta}}_{2}} = {{{- K_{2}}\eta_{1}} + {{ɛ\delta}\left( {x,\overset{\sim}{x},u} \right)}}}\end{matrix} \right\} & (12)\end{matrix}$

As the parameter ε is much smaller than 1, the effects of model error δon the estimation errors η₁, η₂ can be made small enough by equation(12). Thus by using the high-gain observer for a model which hasinjection pressure as a state variable, the above requirement (B)“High-precision detection” for a pressure detecting means(paragraph{0052}) is satisfied.

When the effects of the model error δ on the estimation errors η₁, η₂are neglected, equation (12) is rewritten as equation (13).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 13} \right\} & \; \\{\begin{bmatrix}{\overset{.}{\eta}}_{1} \\{\overset{.}{\eta}}_{2}\end{bmatrix} = {{{\frac{1}{ɛ}\begin{bmatrix}{- K_{1}} & 1 \\{- K_{2}} & 0\end{bmatrix}}\begin{bmatrix}\eta_{1} \\\eta_{2}\end{bmatrix}} = {\frac{1}{ɛ}{A\begin{bmatrix}\eta_{1} \\\eta_{2}\end{bmatrix}}}}} & (13) \\{A = \begin{bmatrix}{- K_{1}} & 1 \\{- K_{2}} & 0\end{bmatrix}} & (14)\end{matrix}$

When K₁, K₂ are decided so that conjugate complex eigenvalues λ₁, λ ₁ ofmatrix A have a negative real part, that is, Re(λ₁)=Re( λ ₁)<0, theestimate errors η₁, η₂ are given by equation (15) with initial valuesη₁₀, η₂₀.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 13} \right\} & \; \\\left. \begin{matrix}{{\eta_{1}(t)} = {{\exp \left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}\left( {{{C_{1}(t)}\eta_{10}} + {{C_{2}(t)}\eta_{20}}} \right)}} \\{{\eta_{2}(t)} = {{\exp \left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}\left( {{{C_{3}(t)}\eta_{10}} + {{C_{4}(t)}\eta_{20}}} \right)}}\end{matrix} \right\} & (15)\end{matrix}$

where t: Time variable, C₁(t)˜C₄ (t): Sinusoidal components withconstant amplitudes and constant frequency decided by K₁, K₂. AsRe(λ₁)<0 and ε is much smaller than 1, equation (15) reveals that thetime responses η₁(t), η₂(t) of estimation errors tend to zero rapidly.In other words, by using high-gain observer equation (6), the aboverequirement (A) “Detection with small time-lag” for a pressure detectingmeans (paragraph{0052}) can be satisfied.

Although estimates {circumflex over (x)}₁, {circumflex over (x)}₂ of allstate variables are obtained by equation (6), it is sufficient to getonly the estimate {circumflex over (x)}₂ because {circumflex over (x)}₁is detected as output y. Then the high-gain observer is given byequation (16).

{Math. 14}

{dot over ({circumflex over (x)} ₂ =−H{circumflex over (x)} ₂ +H{dotover (y)}+φ ₀({circumflex over (x)} ₂ , y, u)   (16)

where H: Gain constant of the high-gain observer which is larger than 1.As time-derivative term of output y is included in the right-hand sideof equation (16), equation (16) cannot be used as a computing equationby itself. But it can be shown that the high-gain observer by equation(16) satisfies the above two requirements (A) and (B) (paragraph{0052}).Equation (17) is given from the third equation in equation (5).

{Math. 15}

{dot over (y)}={dot over (x)}₁=x₂   (17)

Equation (18) is given by using equations (16) and (17).

{Math. 16}

{dot over ({circumflex over (x)} ₂ =−H{circumflex over (x)} ₂ +Hx₂+φ₀({circumflex over (x)} ₂ , y, u)   (18)

By using the second equation of equation (5), equation (19) is givenfrom equation (18).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 17} \right\} & \; \\{{\overset{\overset{.}{\sim}}{x}}_{2} = {{{- H}{\overset{\sim}{x}}_{2}} + {\delta \left( {x,{\overset{\sim}{x}}_{2},y,u} \right)}}} & (19) \\\left. \begin{matrix}{{\overset{\sim}{x}}_{2} = {x_{2} - {\hat{x}}_{2}}} \\{{\delta \left( {x,{\overset{\sim}{x}}_{2},y,u} \right)} = {{\varphi \left( {x,u} \right)} - {\varphi_{0}\left( {{\hat{x}}_{2},y,u} \right)}}}\end{matrix} \right\} & (20)\end{matrix}$

Gain constant H is given by equation (21) by introducing a positiveparameter 6 much smaller than 1.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 18} \right\} & \; \\{H = {\frac{K}{ɛ}\mspace{14mu} \left( {K > 0} \right)}} & (21)\end{matrix}$

By using equation (21), equation (19) is rewritten as equation (22).

{Math. 19}

ε{dot over ({tilde over (x)} ₂ =−K{tilde over (x)} ₂+εδ(x, {tilde over(x)} ₂ , y, u)   (22)

As ε is much smaller than 1, the effect of model error δ on theestimation error {tilde over (x)}₂ can be made small enough fromequation (22). Therefore by using the high-gain observer for a modelwhich has injection pressure as a state variable, the above requirement(B) “High-precision detection” for a pressure detecting means(paragraph{0052}) can be satisfied.

When the effect of model error δ on the estimation error {tilde over(x)}₂ is neglected, equation (22) is rewritten as equation (23).

{Math. 20}

ε{dot over ({tilde over (x)}₂=−K{tilde over (x)}₂   (23)

The estimation error {tilde over (x)}₂ is given by equation (24) fromequation (23).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 21} \right\} & \; \\{{{\overset{\sim}{x}}_{2}(t)} = {{\exp \left( {{- \frac{K}{ɛ}}t} \right)}{\overset{\sim}{x}}_{20}}} & (24)\end{matrix}$

where {tilde over (x)}₂₀ : Initial value of {tilde over (x)}₂. As ε ismuch smaller than 1, equation (24) reveals that the time response {tildeover (x)}₂(t) of estimation error tends to zero rapidly. In other words,by using high-gain observer equation (16), the above requirement (A)“Detection with small time-lag” for a pressure detecting means(paragraph{0052}) can be satisfied. As in equation (16) the minimumnumber of state variables to be estimated are included and themeasurable state variables are excluded, equation (16) is called by areduced-order high-gain observer because the order of observer equation(16) is lower than that of observer equation (6).

Then a procedure to modify equation (16) is shown so that thetime-derivative term of output y is not appeared. A new variable ŵ isgiven by equation (25).

{Math. 22}

ŵ={circumflex over (x)} ₂−Hy   (25)

By using equation (25), equation (16) is rewritten as equation (26).

{Math. 23}

{dot over (ŵ)}=−H(ŵ+Hy)+φ₀(ŵ, y, u)   (26)

Variable ŵ is calculated by equation (26) and estimate {circumflex over(x)}₂ is obtained by equation (27).

{Math. 24}

{circumflex over (x)} ₂ =ŵ+Hy   (27)

Procedures of applying a high-gain observer to a model of electric-motordriven injection molding machines which has injection pressure as astate variable are described in detail in Example 1 and Example 2 to behereinafter described.

ADVANTAGEOUS EFFECTS OF INVENTION

By applying a high-gain observer to a model of electric-motor driveninjection molding machines which has injection pressure as a statevariable, a high-precision pressure detection with small time-lagbecomes possible without using a pressure detector. By using thehigh-gain observer the two requirements (1) and (2) described inparagraph {0008} for controlling pressure of electric-motor driveninjection molding machines can be satisfied and also the fourdisadvantages described in Background Art (paragraph{0046}) can beavoided.

BRIEF DESCRIPTION OF DRAWINGS

{FIG. 1} FIG. 1 is an explanation drawing of working example 1 whichshows a block diagram of a controller for an electric-motor driveninjection molding machine according to an embodiment of the presentinvention.

{FIG. 2} FIG. 2 is an explanation drawing of working example 2 whichshows a block diagram of a controller for an electric-motor driveninjection molding machine according to an embodiment of the presentinvention.

{FIG. 3} FIG. 3 is a view which shows an existing injection and pressureapplication mechanism of an electric-motor driven injection moldingmachine.

{FIG. 4} FIG. 4 is an explanation drawing which shows a time schedule ofmold good manufacturing.

{FIG. 5} FIG. 5 is an explanation drawing which shows a block diagram ofan existing controller for an electric-motor driven injection moldingmachine.

{FIG. 6} FIG. 6 is a view which shows an injection and pressureapplication mechanism of an electric-motor driven injection moldingmachine according to an embodiment of the present invention.

{FIG. 7} FIG. 7 is an explanation drawing of working example 1 whichshows computer simulation results of injection pressure estimation bythe high-gain observer according to an embodiment of the presentinvention.

{FIG. 8} FIG. 8 is an explanation drawing of working example 1 whichshows computer simulation results of injection velocity estimation bythe high-gain observer according to an embodiment of the presentinvention.

{FIG. 9} FIG. 9 is an explanation drawing of working example 2 whichshows computer simulation results of injection pressure estimation bythe high-gain observer according to an embodiment of the presentinvention.

DESCRIPTION OF EMBODIMENT

Hereinafter, the embodiment of the present invention on the controllerof electric-motor driven injection molding machines is described basedon the drawings.

Example 1

FIG. 6 is a view which shows an injection and pressure applicationmechanism without using a pressure detector. As the mechanism in FIG. 6consists of the parts with the same reference signs as in FIG. 3 excepta pressure detector 10, explanations of FIG. 6 are replaced by those ofFIG. 3 described in Background Art (paragraph{0004}˜{0006}).

FIG. 1 is an example of a controller of an electric-motor driveninjection molding machine using a high-gain observer as an injectionpressure detecting means according to an embodiment of the presentinvention and shows a block diagram of the controller. The controllerconsists of an injection controller 20 which contains a high-gainobserver 31 and a motor controller 40.

The injection controller 20 is explained as follows. The injectioncontroller 20 executes a control algorithm at a constant time intervalΔt and feeds a discrete-time control demand to the motor controller 40.The injection controller 20 consists of an injection velocity settingdevice 21, a transducer 22, a pulse generator 23, an injection pressuresetting device 26, a subtracter 27, a pressure controller 28, a D/Aconverter 29, an A/D converter 30 and a high-gain observer 31.

The injection velocity setting device 21 feeds a time sequence ofinjection velocity command V_(i)* to the transducer 22. The transducer22 calculates screw displacement command Δx_(v)* for the screw 9 whichhas to move during the time interval Δt by the following equation (28).

{Math. 25}

Δx_(v)* =V_(i)*Δt   (26)

The command Δx_(v)* is fed to the pulse generator 23 which feeds a pulsetrain 24 corresponding to the command Δx_(v)*.The pulse train 24 is fedto a pulse counter 41 in the motor controller 40.

The injection pressure setting device 26 feeds a time sequence ofinjection pressure command P_(i)* to the subtracter 27. Motor currentdemand i* in the motor controller 40 is fed to the high-gain observer 31through the A/D converter 30 in the injection controller 20. The screwposition signal x which is fed by a pulse counter 44 in the motorcontroller 40 is fed to the high-gain observer 31. The high-gainobserver 31 executes discrete-time arithmetic expressions which areobtained from a mathematical model of an injection mechanism and outputsan estimate of injection pressure {circumflex over (P)}_(i) and anestimate of injection velocity {circumflex over (v)} not shown in FIG. 1by using the input signals x and i*.

The estimate {circumflex over (P)}_(i) is fed to the subtracter 27. Thesubtracter 27 calculates a pressure control deviation ΔP_(i) frominjection pressure command P_(i)* by equation (29).

{Math. 26}

ΔP _(i) =P _(i) *−{circumflex over (P)} _(i)   (29)

The subtracter 27 feeds ΔP_(i) to the pressure controller 28.

The pressure controller 28 calculates a motor current demand i_(p)* fromΔP_(i) by using PID control algorithm and feeds the demand i_(p)* to themotor controller 40 through the D/A converter 29.

The motor controller 40 is explained as follows. The motor controller 40consists of pulse counters 41 and 44, an A/D converter 42, a comparator43, subtracters 45 and 48, a position controller 46, a differentiator47, a velocity controller 49 and a PWM device 50. The motor controller40 is connected to the servomotor 3 equipped with the rotary encoder 12.The motor current demand i_(p)* is fed to the comparator 43 through theA/D converter 42.

The pulse counter 41 accumulates the pulse train 24 from the injectioncontroller 20 and obtains screw position demand x* and outputs thedemand x* to the subtracter 45. The pulse counter 44 accumulates thepulse train from the rotary encoder 12 and obtains the actual screwposition x and feeds the position x to the subtracter 45.

The subtracter 45 calculates a position control deviation (x*−x) andfeeds the position deviation to the position controller 46. The positioncontroller 46 calculates velocity demand v* by the following equation(30) and feeds the demand v* to the subtracter 48.

{Math. 27}

v*=K _(P)(x*−x)   (30)

where Kp is a proportional gain constant of the position controller 46.The rotary encoder 12 feeds the pulse train to the differentiator 47 andto the pulse counter 44. The differentiator 47 detects an actual screwvelocity v and feeds the velocity v to the subtracter 48.

-   -   The subtracter 48 calculates a velocity control deviation (v*−v)        and feeds the deviation to the velocity controller 49. The        velocity controller 49 calculates a motor current demand i_(v)*        by the following equation (31) and feeds the demand i_(v)* to        the comparator 43.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 28} \right\} & \; \\{i_{v}^{*} = {{K_{Pv}\left( {v^{*} - v} \right)} + {\frac{K_{Pv}}{T_{Iv}}{\int{\left( {v^{*} - v} \right){t}}}}}} & (31)\end{matrix}$

where K_(Pv) and T_(Iv) are a proportional gain constant and an integraltime constant of the velocity controller 49, respectively.

The comparator 43 to which motor current demands i_(v)* and i_(p)* aregiven from the velocity controller 49 and the pressure controller 28,respectively, selects the lower current demand i* of i_(v)* and andfeeds the demand i* to the PWM device 50. The PWM device 50 appliesthree-phase voltage to the servomotor 3 so that the servomotor 3 isdriven by the motor current demand i*.

That the above two requirements (1) and (2) (paragraph{0008}) for thecontroller of electric-motor driven injection molding machines arerealized by the comparator 43, is already described in detail inBackground Art (paragraph{0032}˜{0035}).

The high-gain observer 31 outputs estimate A of injection pressure andestimate {circumflex over (v)} of injection velocity by using screwposition signal x and motor current demand signal i*. The mathematicalmodel of an injection mechanism shown in FIG. 6 is derived as follows,which is necessary to design the high-gain observer 31. A motionequation of the motor 3 axis is given by equation (32).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 29} \right\} & \; \\{{\left( {J_{M} + J_{G\; 1}} \right)\frac{\omega_{m}}{t}} = {T_{M} - {r_{1}F}}} & (32)\end{matrix}$

where J_(M): Moment of inertia of motor itself, J_(G1): Moment ofinertia of motor-side gear, ω_(m): Angular velocity of motor, T_(M):Motor torque, r₁: Radius of motor-side gear, F: Transmission force ofreduction gear, t: Time variable. A motion equation of the ball screw 5axis is given by equation (33).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 30} \right\} & \; \\{{\left( {J_{S} + J_{G\; 2}} \right)\frac{\omega_{s}}{t}} = {{r_{2}F} - T_{a}}} & (33)\end{matrix}$

where J_(S): Moment of inertia of ball screw axis, J_(G2): Moment ofinertia of load-side gear, ω_(s): Angular velocity of ball screw axis,r₂: Radius of load-side gear, T_(a): Ball screw drive torque. A motionequation of the moving part 8 is given by equations (34) and (35).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 31} \right\} & \; \\{{\frac{W}{g}\frac{v}{t}} = {F_{a} - F_{L} - {\mu \; W\frac{v}{v}}}} & (34) \\{\frac{x}{t} = v} & (35)\end{matrix}$

where W: Weight of the moving part 8, g: Gravity acceleration, v:Velocity of the moving part (the screw), x: Screw position (initialposition x=0), F_(a): Drive force of the ball screw, F_(L): Appliedforce by polymer to the screw, μ: Friction coefficient at the slider. Arelation between ball screw drive force F_(a) and ball screw drivetorque T_(a), is given by equation (36).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 32} \right\} & \; \\{T_{a} = {\frac{l}{2\pi}\frac{1}{\eta}F_{a}}} & (36)\end{matrix}$

where l: Ball screw lead, η: Ball screw efficiency. Equations among v,ω_(s) and ω_(m) are given by equation (37).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 33} \right\} & \; \\{v = {{\frac{l}{2\pi}\omega_{s}} = {\frac{l}{2\pi}\frac{r_{1}}{r_{2}}\omega_{m}}}} & (37)\end{matrix}$

Applied force to the screw F_(L) is given by equation (38).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 34} \right\} & \; \\{F_{L} = {{A_{s}P_{i}} + {C_{mt}\frac{v}{v}{v}^{\alpha}}}} & (38)\end{matrix}$

where A_(s): Screw section area, P_(i): Injection pressure which meanspolymer pressure at the end of a barrel, C_(mt): Friction coefficientbetween the screw and the barrel surface, α: Velocity power coefficient.A dynamic equation of injection pressure P_(i) is given by equations(39) and (40).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 35} \right\} & \; \\{{\frac{V_{i}}{\beta}\frac{P_{i}}{t}} = {{A_{s}v} - Q_{in}}} & (39) \\{V_{i} = {V_{i\; 0} - {A_{s}x}}} & (40)\end{matrix}$

where V_(i): Polymer volume at the end of a barrel, V_(i0): Initialvolume of V_(i), Q_(in): Injected rate of polymer, β: Bulk modulus ofpolymer. The characteristics of the servomotor 3 is given by equation(41).

{Math. 36}

T_(M)=K_(T)i_(m)   (41)

where K_(T): Motor torque coefficient, i_(m): Motor current. By usingequations (32), (33) and (37) and deleting ω_(s) and F, equation (42) isderived.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 37} \right\} & \; \\{{\left\{ {J_{M} + J_{G\; 1} + {\left( {J_{S} + J_{G\; 2}} \right)\left( \frac{r_{1}}{r_{2}} \right)}} \right\} \frac{\omega_{m}}{t}} = {T_{M} - {\frac{r_{1}}{r_{2}}T_{a}}}} & (42)\end{matrix}$

By using equations (34), (36), (37) and (42) and deleting T_(a) andF_(a), equation (43) is derived.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 38} \right\} & \; \\{{J_{eq}\frac{\omega_{m}}{t}} = {T_{M} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\left( {F_{L} + {\mu \; W\frac{v}{v}}} \right)}}} & (43) \\{J_{eq} = {J_{M} + J_{G\; 1} + {\left( {J_{S} + J_{G\; 1}} \right)\left( \frac{r_{1}}{r_{2}} \right)^{2}} + {\frac{W}{g}\left( \frac{r_{1}}{r_{2}} \right)^{2}\left( \frac{l}{2\pi} \right)^{2}\frac{1}{\eta}}}} & (44)\end{matrix}$

where J_(eq): Reduced moment of inertia at motor axis. Equation (43) isthe motion equation of total injection molding mechanism converted tothe motor axis. From equations (35) and (37), equation (45) is derived.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 39} \right\} & \; \\{\frac{x}{t} = {\frac{r_{1}}{r_{2}}\frac{l}{2\pi}\omega_{m}}} & (45)\end{matrix}$

From equations (38), (41) and (43), the motion equation of the totalmechanism is given by equation (46).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 40} \right\} & \; \\{{J_{eq}\frac{\omega_{m}}{t}} = {{K_{T}i_{m}} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\left\{ {{A_{s}P_{i}} + {C_{mt}\frac{v}{v}{v}^{\alpha}} + {\mu \; W\frac{v}{v}}} \right\}}}} & (46)\end{matrix}$

Equation (40) is rewritten as equation (47).

{Math. 41}

V _(i) =A _(s)(x _(max) −x)   (47)

where x_(max): Maximum screw stroke. By using equations (37) and (47),equation (39) is rewritten as equation (48).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 42} \right\} & \; \\{{\frac{A_{s}\left( {x_{\max} - x} \right)}{\beta}\frac{P_{i}}{t}} = {{A_{s}\frac{l}{2\pi}\frac{r_{1}}{r_{2}}\omega_{m}} - Q_{in}}} & (48)\end{matrix}$

The variables in the above equations are made dimensionless. By usingdimensionless variables, equation (45) is rewritten as equation (49).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 43} \right\} & \; \\{{\frac{}{t}\left\lbrack \frac{x}{x_{\max}} \right\rbrack} = {{\frac{l}{2\pi}\frac{r_{1}}{r_{2}}{\frac{\omega_{\max}}{x_{\max}}\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}} = {\frac{v_{\max}}{x_{\max}}\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}}} & (49)\end{matrix}$

where ω_(max): Motor rating speed, v_(max): Maximum injection velocity.

By using dimensionless variables, equation (46) is rewritten as equation(50).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 44} \right\} & \; \\{{J_{eq}\omega_{\max}{\frac{}{t}\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}} = {{K_{T}{i_{\max}\left\lbrack \frac{i_{m}}{i_{\max}} \right\rbrack}} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}A_{s}{P_{\max}\left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack}} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\frac{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack }\left\{ {{C_{mt}v_{\max}^{\alpha}{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack }^{\alpha}} + {\mu \; W}} \right\}}}} & (50)\end{matrix}$

where i_(max): Motor current rating, P_(max): Maximum injectionpressure. In deriving equation (50), equation (51) is used.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 45} \right\} & \; \\{\left\lbrack \frac{v}{v_{\max}} \right\rbrack = \left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack} & (51)\end{matrix}$

Equation (50) is rewritten as equation (52).

$\begin{matrix}{\mspace{85mu} \left\{ {{Math}.\mspace{14mu} 46} \right\}} & \; \\{{\frac{}{t}\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack} = {{\frac{T_{Mmax}}{J_{eq}\omega_{\max}}\left\lbrack \frac{i_{m}}{i_{\max}} \right\rbrack} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}{\frac{A_{s}P_{\max}}{J_{eq}\omega_{\max}}\left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack}} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\frac{1}{J_{eq}\omega_{\max}}\frac{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack }\left\{ {{C_{mt}v_{\max}^{\alpha}{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack }^{\alpha}} + {\mu \; W}} \right\}}}} & (52)\end{matrix}$

where T_(Mmax)=K_(T)i_(max): Motor rating torque.

By using dimensionless variables, equation (48) is rewritten as equation(53).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 47} \right\}} & \; \\{{\frac{1}{\beta}A_{s}x_{\max}P_{\max}\left\{ {1 - \left\lbrack \frac{x}{x_{\max}} \right\rbrack} \right\} {\frac{}{t}\left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack}} = {A_{s}v_{\max}\left\{ {\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack - \left\lbrack \frac{Q_{i\; n}}{Q_{\max}} \right\rbrack} \right\}}} & (53)\end{matrix}$

where Q_(max)=A_(s)v_(max): Maximum injection rate. Equation (53) isrewritten as equation (54).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 48} \right\} & \; \\{{\frac{}{t}\left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack} = {\frac{\beta}{1 - \left\lbrack \frac{x}{x_{\max}} \right\rbrack}\frac{v_{\max}}{x_{\max}P_{\max}}\left\{ {\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack - \left\lbrack \frac{Q_{in}}{Q_{\max}} \right\rbrack} \right\}}} & (54)\end{matrix}$

In general dimensionless injection rate [Q_(in)/Q_(max)] is a functionof dimensionless injection pressure [P_(i)/P_(max)] given by equation(55). The function (55) is decided by a nozzle shape of a barrel, anentrance shape of the mold, a cavity shape and a polymercharacteristics.

$\begin{matrix}\left\lbrack {{Math}.\mspace{14mu} 49} \right\} & \; \\{\left\lbrack \frac{Q_{in}}{Q_{\max}} \right\rbrack = {f\left( \left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack \right)}} & (55)\end{matrix}$

The mathematical model necessary for designing the high-gain observer 31is given by equations (56), (57) and (58) by using equations (49), (52),(54) and (55).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 50} \right\}} & \; \\{\mspace{79mu} {{\frac{}{t}\left\lbrack \frac{x}{x_{\max}} \right\rbrack} = {\frac{v_{\max}}{x_{\max}}\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}}} & (56) \\{{\frac{}{t}\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack} = {{\frac{T_{Mmax}}{J_{eq}\omega_{\max}}\left\lbrack \frac{i_{m}}{i_{\max}} \right\rbrack} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}{\frac{A_{s}P_{\max}}{J_{eq}\omega_{\max}}\left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack}} - {\frac{l}{2\pi}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\frac{1}{J_{eq}\omega_{\max}}\frac{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack}{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack }\left\{ {{C_{mt}v_{\max}^{\alpha}{\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack }^{\alpha}} + {\mu \; W}} \right\}}}} & (57) \\{\mspace{79mu} {{\frac{}{t}\left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack} = {\frac{\beta}{1 - \left\lbrack \frac{x}{x_{\max}} \right\rbrack}\frac{v_{\max}}{x_{\max}P_{\max}}\left\{ {\left\lbrack \frac{\omega_{m}}{\omega_{\max}} \right\rbrack - {f\left( \left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack \right)}} \right\}}}} & (58)\end{matrix}$

When the cavity is filled up with polymer melt, equation (59) issatisfied.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 51} \right\} & \; \\{{f\left( \left\lbrack \frac{P_{i}}{P_{\max}} \right\rbrack \right)} = 0} & (59)\end{matrix}$

The following state variables x₁, x₂ and x₃ defined by equation (60) areintroduced.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 52} \right\} & \; \\{x_{1} = {{\frac{x}{x_{\max}}\mspace{14mu} x_{2}} = {{\frac{\omega_{m}}{\omega_{\max}}\mspace{14mu} x_{3}} = \frac{P_{i}}{P_{\max}}}}} & (60)\end{matrix}$

Input variable u defined by equation (61) is introduced. u ismeasurable. In the design of high-gain observer 31, the actual motorcurrent i_(m) is considered to be equal to motor current demand i*.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 53} \right\} & \; \\{u = \frac{i_{m}}{i_{\max}}} & (61)\end{matrix}$

The state variable x₁ is supposed to be measurable and output variable yis defined by equation (62).

{Math. 54}

y=x₁   (62)

The state equations representing equations (56), (57), (58) and (62) aregiven by equations (63) and (64).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 55} \right\}} & \; \\\left. \mspace{79mu} \begin{matrix}{{\overset{.}{x}}_{1} = {a\; x_{2}}} \\{{\overset{.}{x}}_{2} = {{bx}_{3} + {\chi \left( x_{2} \right)} + {cu}}} \\{{\overset{.}{x}}_{3} = {{\frac{d}{1 - x_{1}}\left\{ {x_{2} - {f\left( x_{3} \right)}} \right\}} = {\psi (x)}}}\end{matrix} \right\} & (63) \\{\mspace{79mu} {y = x_{1}}} & (64) \\{\mspace{79mu} {x = {{\begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}\mspace{14mu} {\chi \left( x_{2} \right)}} = {e\; \frac{x_{2}}{x_{2}}\left( {{h{x_{2}}^{\alpha}} + p} \right)}}}} & (65) \\\left. \begin{matrix}{a = \frac{v_{\max}}{x_{\max}}} & {b = {{- \frac{l}{2\pi}}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\frac{A_{s}P_{\max}}{J_{eq}\omega_{\max}}}} & {c = \frac{T_{Mmax}}{J_{eq}\omega_{\max}}} & \; \\{d = \frac{\beta \; v_{\max}}{x_{\max}P_{\max}}} & {e = {{- \frac{l}{2\pi}}\frac{1}{\eta}\frac{r_{1}}{r_{2}}\frac{1}{J_{eq}\omega_{\max}}}} & {h = {C_{mt}v_{\max}^{\alpha}}} & {p = {\mu \; W}}\end{matrix} \right\} & (66)\end{matrix}$

where x(x₂) and ψ(x) are nonlinear functions. Equations (63) and (64)are rewritten as equations (67) and (68) by using a vector x in theabove equation (65).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 56} \right\} & \; \\{\overset{.}{x} = {{{Ax} + {\begin{bmatrix}0 \\{{\chi \left( x_{2} \right)} + {cu}} \\{\psi (x)}\end{bmatrix}\mspace{14mu} A}} = \begin{bmatrix}0 & a & 0 \\0 & 0 & b \\0 & 0 & 0\end{bmatrix}}} & (67) \\{y = {\begin{bmatrix}1 & 0 & 0\end{bmatrix}x}} & (68)\end{matrix}$

New state variables X₁ and X₂ are introduced by equation (69).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 57} \right\} & \; \\{X_{1} = {{x_{1}\mspace{14mu} X_{2}} = {{\begin{bmatrix}x_{2} \\x_{3}\end{bmatrix}\mspace{14mu} X} = \begin{bmatrix}X_{1} \\X_{2}\end{bmatrix}}}} & (69)\end{matrix}$

Equations (67) and (68) are rewritten as equations (70) and (71) byusing equation (69).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 58} \right\}} & \; \\{\mspace{79mu} {\begin{bmatrix}{\overset{.}{X}}_{1} \\{\overset{.}{X}}_{2}\end{bmatrix} = {{\begin{bmatrix}A_{11} & A_{12} \\A_{21} & A_{22}\end{bmatrix}\begin{bmatrix}X_{1} \\X_{2}\end{bmatrix}} + {\begin{bmatrix}B_{1} \\B_{2}\end{bmatrix}{\varphi \left( {X,u} \right)}}}}} & (70) \\{\mspace{79mu} {y = X_{1}}} & (71) \\\left. \begin{matrix}{A_{11} = 0} & {A_{12} = \begin{bmatrix}a & 0\end{bmatrix}} & {A_{21} = \begin{bmatrix}0 \\0\end{bmatrix}} & {A_{22} = \begin{bmatrix}0 & b \\0 & 0\end{bmatrix}} \\{B_{1} = \begin{bmatrix}0 & 0\end{bmatrix}} & {B_{2} = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}} & \; & {{\varphi \left( {X,u} \right)} = \begin{bmatrix}{{\chi \left( X_{2} \right)} + {cu}} \\{\psi (X)}\end{bmatrix}}\end{matrix} \right\} & (72)\end{matrix}$

Equation (70) is rewritten as equation (73).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 59} \right\} & \; \\\left. \begin{matrix}{{\overset{.}{X}}_{1} = {A_{12}X_{2}}} \\{{\overset{.}{X}}_{2} = {{A_{22}X_{2}} + {\varphi \left( {X,u} \right)}}}\end{matrix} \right\} & (73)\end{matrix}$

As state variable X₁=x₁ is measurable, it is not necessary to estimatestate variable X₁. Therefore the high-gain observer 31 outputs theestimate of state variable {circumflex over (X)}₂ by using the screwposition signal X₁=x₁ and the motor current demand u. The estimate{circumflex over (X)}₂ is given by equation (74) (non patent literatureNPL 2).

{Math. 60}

{dot over ({circumflex over (X)} ₂=(A ₂₂ −KA ₁₂){circumflex over (X)} ₂+K{dot over (y)}+φ ₀({circumflex over (X)} ₂ , y, u)   (74)

where K: Gain constant matrix of the high-gain observer 31,φ₀({circumflex over (X)}₂, y, u): Nominal function of φ(X, u). Equation(74) is rewritten by equation (75).

{Math. 61}

{dot over ({circumflex over (X)} ₂ −K{dot over (y)}=(A ₂₂ −KA₁₂){circumflex over (X)} ₂+φ₀({circumflex over (X)} ₂ , y, u)   (75)

A new variable ŵ is introduced by equation (76).

{Math. 62}

ŵ={circumflex over (X)} ₂ −Ky   (76)

The estimate {circumflex over (X)}₂ is given by equations (77) and (78)by using equations (75) and (76).

{Math. 63}

{dot over (ŵ)}=(A ₂₂ −KA ₁₂)(ŵ+Ky)+φ₀(ŵ, y, u)   (77)

{circumflex over (X)} ₂ =ŵ+Ky   (78)

positive parameter E much smaller than 1 is introduced and the gainmatrix K is given by equation (79).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 64} \right\} & \; \\{K = \begin{bmatrix}{K_{1}/ɛ} \\{K_{2}/ɛ^{2}}\end{bmatrix}} & (79)\end{matrix}$

Equation (77) is rewritten as equation (80) from equations (72) and(79).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 65} \right\}} & \; \\{\begin{bmatrix}{\overset{\overset{.}{\hat{}}}{\omega}}_{1} \\{\overset{\overset{.}{\hat{}}}{\omega}}_{2}\end{bmatrix} = {{\begin{bmatrix}{{- {aK}_{1}}/ɛ} & b \\{{- {aK}_{2}}/ɛ^{2}} & 0\end{bmatrix}\begin{bmatrix}{\hat{\omega}}_{1} \\{\hat{\omega}}_{2}\end{bmatrix}} + {\begin{bmatrix}{\left( {{- {aK}_{1}^{2}} + {bK}_{2}} \right)/ɛ^{2}} \\{{- {aK}_{1}}{K_{2}/ɛ^{3}}}\end{bmatrix}y} + {\quad\begin{bmatrix}{{\chi_{0}\left( {{\hat{\omega}}_{1},y} \right)} + {cu}} \\{\psi_{0}\left( {\hat{\omega},y} \right)}\end{bmatrix}}}} & (80)\end{matrix}$

where x₀(ŵ₁, y) and ψ₀(ŵ, y) are nominal functions of x(X₂) and ψ(X),respectively.

New variables {circumflex over (η)}₁, {circumflex over (η)}₂ are givenby equation (81).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 66} \right\} & \; \\{\hat{\eta} = {\begin{bmatrix}{\hat{n}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix} = {{\begin{bmatrix}ɛ & 0 \\0 & ɛ^{2}\end{bmatrix}\begin{bmatrix}{\hat{\omega}}_{1} \\{\hat{\omega}}_{2}\end{bmatrix}} = {{{D\begin{bmatrix}{\hat{\omega}}_{1} \\{\hat{\omega}}_{2}\end{bmatrix}}\mspace{14mu} D} = \begin{bmatrix}ɛ & 0 \\0 & ɛ^{2}\end{bmatrix}}}}} & (81)\end{matrix}$

Equation (80) is rewritten as equation (82) by using equation (81).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 67} \right\} & \; \\{{\begin{bmatrix}{\overset{\overset{.}{\hat{}}}{\eta}}_{1} \\{\overset{\overset{.}{\hat{}}}{\eta}}_{2}\end{bmatrix} = {{\frac{1}{ɛ}\left( {A_{22} - {K_{0}A_{12}}} \right)\left\{ {\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix} + K_{0y}} \right\}} + \begin{bmatrix}{{{ɛ\chi}_{0}\left( {{\hat{n}}_{1},y} \right)} + {ɛ\; {cu}}} \\{ɛ^{2}{\psi_{0}\left( {\hat{\eta},y} \right)}}\end{bmatrix}}}{K_{0} = \begin{bmatrix}K_{1} \\K_{2}\end{bmatrix}}} & (82)\end{matrix}$

Equation (83) is given from equation (81).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 68} \right\} & \; \\{\begin{bmatrix}{\hat{\omega}}_{1} \\{\hat{\omega}}_{2}\end{bmatrix} = {D^{- 1}\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix}}} & (83)\end{matrix}$

Equation (79) is rewritten as equation (84).

{Math. 69}

K=D⁻¹K₀   (84)

By using equations (83) and (84), equation (78) is rewritten as equation(85).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 70} \right\} & \; \\{{\hat{X}}_{2} = {\begin{bmatrix}{\hat{x}}_{2} \\{\hat{x}}_{3}\end{bmatrix} = {{{D^{- 1}\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix}} + {D^{- 1}K_{0}y}} = \begin{bmatrix}{\left( {{\hat{\eta}}_{1} + {K_{1}y}} \right)/ɛ} \\{\left( {{\hat{\eta}}_{2} + {K_{2}y}} \right)/ɛ^{2}}\end{bmatrix}}}} & (85)\end{matrix}$

Thus the estimates of state variables {circumflex over (x)}₂,{circumflex over (x)}₃ are obtained by the high-gain observer 31. Fromequations (82) and (85), the calculation procedures are given byequations (86) and (87).

(1) Calculation procedure 1

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 71} \right\} & \; \\{\begin{bmatrix}{\overset{\overset{.}{\hat{}}}{\eta}}_{1} \\{\overset{\overset{.}{\hat{}}}{\eta}}_{2}\end{bmatrix} = {{\frac{1}{ɛ}\left( {A_{22} - {K_{0}A_{12}}} \right)\left\{ {\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix} + {K_{0}y}} \right\}} + \begin{bmatrix}{{{ɛ\chi}_{0}\left( {{\hat{\eta}}_{1},y} \right)} + {ɛ\; {cu}}} \\{ɛ^{2}{\psi_{0}\left( {{\hat{\eta}}_{1},{\hat{\eta}}_{2},y} \right)}}\end{bmatrix}}} & (86)\end{matrix}$

(2) Calculation procedure 2

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 72} \right\} & \; \\{\begin{bmatrix}{\hat{x}}_{2} \\{\hat{x}}_{3}\end{bmatrix} = {{{D^{- 1}\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix}} + {D^{- 1}K_{0}y}} = \begin{bmatrix}{\left( {{\hat{\eta}}_{1} + {K_{1}y}} \right)/ɛ} \\{\left( {{\hat{\eta}}_{2} + {K_{2}y}} \right)/ɛ^{2}}\end{bmatrix}}} & (87)\end{matrix}$

By the calculation procedure 1, the estimates {circumflex over (η)}₁ and{circumflex over (η)}₂ are obtained and by the calculation procedure 2the estimates {circumflex over (x)}₂ and {circumflex over (x)}₃ areobtained.

Then it is shown that the high-gain observer 31 as the pressuredetecting means satisfies the following two requirements (A) and (B)described in Solution to Problem (paragraph { 0052}).

(A) The detection means has very small time-lag.

(B) The detection means is high-precision.

If the nominal functions x₀({circumflex over (η)}₁, y) andψ₀({circumflex over (η)}, y) in equation (86) are replaced with the truebut actually unobtainable functions x(η₁, y) and ψ(η, y), the truevalues η₁, η₂ of the estimates {circumflex over (η)}₁, {circumflex over(η)}₂ may be obtained by equation (88).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 73} \right\} & \; \\{\begin{bmatrix}{\overset{.}{\eta}}_{1} \\{\overset{.}{\eta}}_{2}\end{bmatrix} = {{\frac{1}{ɛ}\left( {A_{22} - {K_{0}A_{12}}} \right)\left\{ {\begin{bmatrix}\eta_{1} \\\eta_{2}\end{bmatrix} + {K_{0}y}} \right\}} + \begin{bmatrix}{{{ɛ\chi}\left( {\eta_{1},y} \right)} + {ɛ\; {cu}}} \\{ɛ^{2}{\psi \left( {\eta_{1},\eta_{2},y} \right)}}\end{bmatrix}}} & (88)\end{matrix}$

Then the estimate errors {tilde over (η)}₁=η₁−{circumflex over (η)}₁ and{tilde over (η)}₂=η₂−{circumflex over (η)}₂ are obtained by equation(89) by using equations (86) and (88).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 74} \right\} & \; \\\left. \begin{matrix}{\begin{bmatrix}{\overset{\overset{.}{\sim}}{\eta}}_{1} \\{\overset{\overset{.}{\sim}}{\eta}}_{2}\end{bmatrix} = {{\frac{1}{ɛ}{\left( {A_{22} - {K_{0}A_{12}}} \right)\begin{bmatrix}{\overset{\sim}{\eta}}_{1} \\{\overset{\sim}{\eta}}_{2}\end{bmatrix}}} + \begin{bmatrix}{{ɛ\delta}_{1}\left( {\eta_{1},{\hat{\eta}}_{1},y} \right)} \\{ɛ^{2}{\delta_{2}\left( {\eta,\hat{\eta},y} \right)}}\end{bmatrix}}} \\{{\delta_{1}\left( {\eta_{1},{\hat{\eta}}_{1},y} \right)} = {{\chi \left( {\eta_{1},y} \right)} - {\chi_{0}\left( {{\hat{\eta}}_{1},y} \right)}}} \\{{\delta_{2}\left( {\eta,\hat{\eta},y} \right)} = {{\psi \left( {\eta,y} \right)} - {\psi_{0}\left( {\hat{\eta},y} \right)}}}\end{matrix} \right\} & (89)\end{matrix}$

As ε is much smaller than 1, the effects of model errors δ₁ and δ₂ onthe estimation errors {tilde over (η)}₁ and {tilde over (η)}₂ can bemade small enough by equation (89). In other words, the high-gainobserver 31 satisfies the above requirement (B) “High-precisiondetection” (paragraph{0052}) for injection pressure estimate {circumflexover (x)}₃ and injection velocity (motor speed) estimate {circumflexover (x)}₂.

When the effects of model errors δ₁ and δ₂ on the estimation errors areneglected in equation (89), equation (89) is rewritten as equation (90).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 75} \right\} & \; \\{\begin{bmatrix}{\overset{\overset{.}{\sim}}{\eta}}_{1} \\{\overset{\overset{.}{\sim}}{\eta}}_{2}\end{bmatrix} = {{\frac{1}{ɛ}{A_{0}\begin{bmatrix}{\hat{\eta}}_{1} \\{\overset{\sim}{\eta}}_{2}\end{bmatrix}}\mspace{14mu} A_{0}} = {A_{22} - {K_{0}A_{12}}}}} & (90)\end{matrix}$

When matrix K₀ is decided so that the real part of conjugate complexeigenvalues λ₁, λ ₁ of matrix A₀ becomes negative, that is, Re(λ₁)=Re( λ₁)<0, the estimate errors {tilde over (η)}₁, {tilde over (η)}₂ are givenby equation (91) with initial values {tilde over (η)}₁₀, {tilde over(η)}₂₀.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 76} \right\} & \; \\\left. \begin{matrix}{{{\overset{\sim}{\eta}}_{1}(t)} = {{\exp \left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}\left( {{{C_{1}(t)}{\overset{\sim}{\eta}}_{10}} + {{C_{2}(t)}{\overset{\sim}{\eta}}_{20}}} \right)}} \\{{{\overset{\sim}{\eta}}_{2}(t)} = {{\exp \left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}\left( {{{C_{3}(t)}{\overset{\sim}{\eta}}_{10}} + {{C_{4}(t)}{\overset{\sim}{\eta}}_{20}}} \right)}}\end{matrix} \right\} & (91)\end{matrix}$

where t: Time variable, C₁(t)˜C₄ (t): Sinusoidal components withconstant amplitudes and constant frequency decided by elements of matrixA₀. As Re(λ₁)<0 and ε is much smaller than 1, equation (91) reveals thatthe time responses {tilde over (η)}₁(t), {tilde over (η)}₂(t) of theestimate errors tend to zero rapidly. In other words, the high-gainobserver 31 satisfies the above requirement (A) “Detection with smalltime-lag” (paragraph{0052}) for injection pressure estimate {circumflexover (x)}₃ and injection velocity (motor speed) estimate {circumflexover (x)}₂.

As the injection controller 20 executes a control algorithm at aconstant time interval Δt, the arithmetic expressions (86) and (87) ofthe high-gain observer 31 are transformed into the discrete-timearithmetic expressions (non patent literature NPL 3, NPL 4).

A new parameter a is introduced and the time interval Δt is expressed byequation (92).

{Math. 77}

Δt=αε  (92)

Discrete-time equivalent of the continuous-time equation (86) can befound by using the standard method of forward rectangular rule whichgives the relation between the Laplace-transform operator s representingtime-derivative operation and z-transform operator z as follows.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 78} \right\} & \; \\{s = {\frac{z - 1}{\Delta \; t} = \frac{z - 1}{\alpha ɛ}}} & (93)\end{matrix}$

By using equation (93), equation (86) is rewritten as equation (94).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 79} \right\} & \; \\{{\frac{z - 1}{\alpha ɛ}\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix}} = {{\frac{1}{ɛ}A_{0}\left\{ {\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2}\end{bmatrix} + K_{0y}} \right\}} + \begin{bmatrix}{{{ɛ\chi}_{0}\left( {{\hat{\eta}}_{1},y} \right)} + {ɛ\; {cu}}} \\{ɛ^{2}{\psi_{0}\left( {{\hat{\eta}}_{1},{\hat{\eta}}_{2},y} \right)}}\end{bmatrix}}} & (94)\end{matrix}$

The discrete-time expression of equation (94) is given by equation (95).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 80} \right\}} & \; \\{{\begin{bmatrix}{{\hat{\eta}}_{1}\left( {k + 1} \right)} \\{{\hat{\eta}}_{2}\left( {k + 1} \right)}\end{bmatrix} - \begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)}\end{bmatrix}} = {{\alpha \; A_{0}\left\{ {\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)}\end{bmatrix} + {K_{0}{y(k)}}} \right\}} + {\alpha \begin{bmatrix}{{ɛ^{2}{\chi_{0}(k)}} + {ɛ^{2}{{cu}(k)}}} \\{ɛ^{3}{\psi_{0}(k)}}\end{bmatrix}}}} & (95) \\{{{\chi_{0}(k)} = \; {e\frac{{\hat{x}}_{2}(k)}{{{\hat{x}}_{2}(k)}}\left( {h{{{\hat{x}}_{2}(k)}^{\alpha}{+ p}}} \right)}}\mspace{76mu} {{\psi_{0}(k)} = {\frac{d}{1 - {y(k)}}\left\{ {{{\hat{x}}_{2}(k)} - {f\left( {{\hat{x}}_{3}(k)} \right)}} \right\}}}} & (96)\end{matrix}$

where {circumflex over (η)}₁(k) , {circumflex over (η)}₂(k): Estimates{circumflex over (η)}₁(t_(k)), {circumflex over (η)}₂(t_(k)) at adiscrete-time t_(k), y(k),u(k): y(t_(k)), u(t_(k)) at a discrete-timet_(k), {circumflex over (x)}₂(k), {circumflex over (x)}₃ (k): Estimates{circumflex over (x)}₂ (t_(k)), {circumflex over (x)}₃ (t_(k)) at adiscrete-time t_(k), x₀(k), ψ₀(k): x₀(t_(k)) , ψ₀(t _(k)) at adiscrete-time t_(k). x₀(k) is given by equation (65) and ψ₀(k) is givenby the third equation of equation (63). Equation (95) is rewritten asequation (97).

$\begin{matrix}{\mspace{79mu} \left\{ {{Math}.\mspace{14mu} 81} \right\}} & \; \\{\begin{bmatrix}{{\hat{\eta}}_{1}\left( {k + 1} \right)} \\{{\hat{\eta}}_{2}\left( {k + 1} \right)}\end{bmatrix} = {{\left( {I_{2} + {\alpha \; A_{0}}} \right)\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)}\end{bmatrix}} + {\alpha \; A_{0}K_{0}{y(k)}} + {\alpha \begin{bmatrix}{{ɛ^{2}{\chi_{0}(k)}} + {ɛ^{2}{{cu}(k)}}} \\{ɛ^{3}{\psi_{0}(k)}}\end{bmatrix}}}} & (97)\end{matrix}$

where I₂: 2×2 unit matrix.

The discrete-time equivalent of equation (87) is given by equation (98).

$\begin{matrix}\left\lbrack {{Math}.\mspace{14mu} 82} \right\rbrack & \; \\{\begin{bmatrix}{{\hat{x}}_{2}(k)} \\{{\hat{x}}_{3}(k)}\end{bmatrix} = {{D^{- 1}\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)}\end{bmatrix}} + {D^{- 1}K_{0}{y(k)}}}} & (98)\end{matrix}$

The high-gain observer 31 obtains injection pressure estimate{circumflex over (x)}₃(k) and injection velocity (motor speed) estimate{circumflex over (x)}₂(k) at a discrete-time t_(k) by executing thearithmetic expressions of equations (97) and (98) at a time interval Δt.The high-gain observer 31 by equations (97) and (98) does not estimatethe measurable state variable x₁(k) (screw position) and estimates onlynecessary state variables x₃(k) and x₂(k) and so is called by areduced-order high-gain observer.

Injection velocity estimate {circumflex over (x)}₂(k) obtained by thehigh-gain observer 31 can be used to monitor the injection velocityoverspeed in the injection controller 20 and this is not shown inFIG. 1. In the motor controller 40, injection velocity estimate{circumflex over (x)}₂(k) can be fed to the subtracter 48 as thevelocity signal v without using the differentiator 47 in FIG. 1.

FIG. 7 shows the results of computer simulation when the high-gainobserver 31 is used for the pressure control of an electric-motor driveninjection molding machine.

The constants of the mathematical model are as follows.

Maximum screw stroke x_(max)=37.2 cm

Maximum injection velocity v_(max)=13.2 cm/sec

Maximum injection pressure P_(max)=17652 N/cm²

Motor rating speed ω_(max)=209.4 rad/sec (2000rpm)

The constants a, b, c and d in equation (63) are expressed in equation(99). In this calculation it is assumed that resistance componentx(x₂)=0.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 83} \right\} & \; \\\left. \begin{matrix}{a = {0.3556\mspace{14mu} \sec^{- 1}}} \\{b = {{- 6.474}\mspace{14mu} \sec^{- 1}}} \\{c = {3.716\mspace{14mu} \sec^{- 1}}} \\{d = {3.626\mspace{14mu} \sec^{- 1}}}\end{matrix} \right\} & (99)\end{matrix}$

In order to realize an arbitrary injection pressure time responseaccording to the setting value P_(set)(t), a hydraulic characteristicsof solenoid operated proportional relief valve is used for thecharacteristics of function ƒ({circumflex over (P)}_(i)/P_(max)), whichdecides the polymer flow into the cavity according to the value of{circumflex over (P)}_(i)(t)/P_(set)(t). The gain matrix K of equation(79) used by the high-gain observer 31 is given by equation (100). Thedata K₁=0.340, K₂=−0.00316, ε=0.015 and Δt=5 msec are used.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 84} \right\} & \; \\{K = \begin{bmatrix}22.67 \\{- 14.04}\end{bmatrix}} & (100)\end{matrix}$

FIG. 7 shows the control performance of injection pressure[P_(i)/P_(max)]. FIG. 7 (a) shows the time response of injectionpressure [P_(i)(t)/P_(max)] when the hitherto known control system inFIG. 5 is used and the pressure detector 10 is used. When the high-gainobserver 31 is supposed to be used under the control system shown inFIG. 5 and to calculate the estimate of injection pressure [{circumflexover (P)}_(i)/P_(max)] by using screw position signal y(t) and motorcurrent signal u(t), FIG. 7 (b) shows the time response of the estimatedinjection pressure [{circumflex over (P)}_(i)(t) /P_(max)]. FIG. 7 (c)shows the time response of the actual injection pressure[P_(i)(t)/P_(max)] when the control system shown in FIG. 1 is used andthe estimated injection pressure [{circumflex over (P)}_(i)(t)/P_(max)]obtained by the high-gain observer 31 is fed to the subtracter 27 as afeedback signal of injection pressure, in other words, the pressuredetector 10 is not used. The time response of estimated injectionpressure [{circumflex over (P)}_(i)(t)/P_(max)] shown in FIG. 7 (b)agrees well with that of actual injection pressure [P_(i)(t)/P_(max)]shown in FIG. 7 (a). The time response of actual injection pressure[P_(i)(t)/P_(max)] shown in FIG. 7 (a) agrees well with that of actualinjection pressure shown in FIG. 7 (c) when using the high-gain observer31. The transfer from the injection process to the pressure applicationprocess is conducted at the screw position [x/x_(max)]=0.55 and the timet₁ shown in FIG. 4 is 2.6 seconds. Thus the high-gain observer 31 canestimate the injection pressure exactly with small time-lag. Theestimate of injection pressure obtained by the high-gain observer 31 canbe used to monitor the injection pressure in the injection process andcan be used as a feedback signal of injection pressure in the pressureapplication process.

FIG. 8 (a) shows the time response of the actual injection velocity[v(t)/v_(max)] when the control system shown in FIG. 1 is used and FIG.8 (b) shows the time response of the estimated injection velocity[{circumflex over (v)}(t)/v_(max)] obtained by the high-gain observer31. Thus the high-gain observer 31 can also estimate the injectionvelocity exactly with small time-lag.

Example 2

FIG. 2 is an example of a controller of an electric-motor driveninjection molding machine using a high-gain observer as an injectionpressure detecting means according to an embodiment of the presentinvention and shows a block diagram of the controller. The controllerconsists of the injection controller 20 which contains a high-gainobserver 32 and the motor controller (servoamplifier) 40.

The injection controller 20 is explained as follows. The injectioncontroller 20 executes a control algorithm at a constant time intervalΔt and feeds a discrete-time control demand to the motor controller 40.The injection controller 20 consists of the parts with the samereference signs as in Example 1 shown in FIG. 1 and a high-gainobservser 32. Descriptions of functions for the parts with the samereference signs as in Example 1 are replaced with those in Example 1(paragraph{0102}˜{0110}).

The high-gain observer 32 is explained as follows. Motor current demandi* in the motor controller 40 is fed to the high-gain observer 32through the A/D converter 30 in the injection controller 20. The screwposition signal x which is fed by the pulse counter 44 in the motorcontroller 40 is fed to the high-gain observer 32. The screw velocitysignal v which is fed by the differentiator 47 in the motor controller40 is fed to the high-gain observer 32. The high-gain observer 32executes discrete-time arithmetic expressions which are obtained from amathematical model of an injection mechanism and outputs estimates{circumflex over (x)}, {circumflex over (v)} and {circumflex over(P)}_(i), of screw position, injection velocity and injection pressure,respectively, by using the input signals x, v and i*.

The motor controller 40 is explained as follows. The motor controller 40consists of the parts with the same reference signs as in Example 1shown in FIG. 1. Descriptions of functions for the parts with the samereference signs as in Example 1 are replaced with those in Example 1(paragraph{0111}˜{0119}).

The state equation of a mathematical model for an injection mechanismshown in FIG. 6 necessary to design the high-gain observer 32 is thesame as equation (67) in Example 1 and is given by equation (101).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 85} \right\} & \; \\{\overset{.}{x} = {{{Ax} + {\begin{bmatrix}0 \\{{\chi \left( x_{2} \right)} + {cu}} \\{\psi (x)}\end{bmatrix}\mspace{14mu} A}} = {{\begin{bmatrix}0 & a & 0 \\0 & 0 & b \\0 & 0 & 0\end{bmatrix}\mspace{14mu} x} = \begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}}}} & (101)\end{matrix}$

The state variables x₁, x₂, x₃ and input variable u are the same asthose given by equations (60) and (61), respectively in Example 1.

The screw position x₁ and injection velocity (motor speed) x₂ aremeasurable state variables and the output variable y is given byequation (102).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 86} \right\} & \; \\{{y = {\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {{\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\end{bmatrix}\begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}} = {Cx}}}}{C = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\end{bmatrix}}} & (102)\end{matrix}$

The high-gain observer 32 estimates all state variables {circumflex over(x)}₁, {circumflex over (x)}₂ and {circumflex over (x)}₃ by usingmeasurable output variables y₁=x₁, y₂=x₂ and input variable u (motorcurrent demand). The estimates {circumflex over (x)}₁, {circumflex over(x)}₂ and {circumflex over (x)}₃ are given by equation (103) (non patentliterature NPL 2).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 87} \right\} & \; \\{{\overset{\overset{.}{\hat{}}}{x} = {{A\hat{x}} + \begin{bmatrix}0 \\{{\chi_{0}\left( y_{2} \right)} + {cu}} \\{\psi_{0}\left( \hat{x} \right)}\end{bmatrix} - {K\left( {{C\hat{x}} - y} \right)}}}{\hat{x} = \begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2} \\{\hat{x}}_{3}\end{bmatrix}}} & (103)\end{matrix}$

where x₀(y₂), ψ₀({circumflex over (x)}): Nominal functions of x(y₂),ψ(x), respectively, K: Gain matrix of the high-gain observer 32.Introducing a positive parameter ε much smaller than 1, K is given byequation (104).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 88} \right\} & \; \\{K = \begin{bmatrix}{K_{11}/ɛ} & K_{12} \\{K_{21}/ɛ^{2}} & {K_{22}/ɛ} \\{K_{31}/ɛ^{3}} & {K_{32}/ɛ_{2}}\end{bmatrix}} & (104)\end{matrix}$

New estimated variable {circumflex over (η)} is introduced by equation(105).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 89} \right\} & \; \\{{\hat{\eta} = {\begin{bmatrix}{\hat{\eta}}_{1} \\{\hat{\eta}}_{2} \\{\hat{\eta}}_{3}\end{bmatrix} = {{\begin{bmatrix}1 & 0 & 0 \\0 & ɛ & 0 \\0 & 0 & ɛ^{2}\end{bmatrix}\begin{bmatrix}{\hat{x}}_{1} \\{\hat{x}}_{2} \\{\hat{x}}_{3}\end{bmatrix}} = {D\hat{x}}}}}{D = \begin{bmatrix}1 & 0 & 0 \\0 & ɛ & 0 \\0 & 0 & ɛ^{2}\end{bmatrix}}} & (105)\end{matrix}$

By using {circumflex over (η)}, equation (103) is rewritten as equations(106) and (107).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 90} \right\} & \; \\{\overset{\overset{.}{\hat{}}}{\eta} = {\frac{1}{ɛ}\left\{ {{A_{0}\hat{\eta}} + {K_{0}\overset{\_}{y}} + \begin{bmatrix}0 \\{{ɛ^{2}{\chi_{0}\left( y_{2} \right)}} + {ɛ^{2}{cu}}} \\{ɛ^{3}{\psi_{0}\left( {{\hat{\eta}}_{3},y} \right)}}\end{bmatrix}} \right\}}} & (106) \\{{A_{0} = {A - {K_{0}C}}}{K_{0} = \begin{bmatrix}K_{11} & K_{12} \\K_{21} & K_{22} \\K_{31} & K_{32}\end{bmatrix}}{\overset{\_}{y} = \begin{bmatrix}y_{1} \\{ɛ\; y_{2}}\end{bmatrix}}} & (107)\end{matrix}$

Thus the estimates of all state variables {circumflex over (x)}₁,{circumflex over (x)}₂ and {circumflex over (x)}₃ are obtained by thehigh-gain observer 32. From equations (105) and (106), the calculationprocedures are given by equations (108) and (109).

(1) Calculation procedure 1

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 91} \right\} & \; \\{\overset{\overset{.}{\hat{}}}{\eta} = {\frac{1}{ɛ}\left\{ {{A_{0}\hat{\eta}} + {K_{0}\overset{\_}{y}} + \begin{bmatrix}0 \\{{ɛ^{2}{\chi_{0}\left( y_{2} \right)}} + {ɛ^{2}{cu}}} \\{ɛ^{3}{\psi_{0}\left( {{\hat{\eta}}_{3},y} \right)}}\end{bmatrix}} \right\}}} & (108)\end{matrix}$

(2) Calculation procedure 2

{Math. 92}

{circumflex over (x)}=D⁻¹{circumflex over (η)}  (109)

By the calculation procedure 1, the estimate {circumflex over (η)} isobtained and by the calculation procedure 2, the estimate {circumflexover (x)} is obtained.

Then it is shown that the high-gain observer 32 as a pressure detectingmeans satisfies the following two requirements (A) and (B) described inSolution to Problem (paragraph { 0052}).

(A) The detection means has very small time-lag.

(B) The detection means is high-precision.

If the nominal functions x₀(y₂), ψ₀({circumflex over (η)}₃, y) inequation (106) are replaced with the true but actually unobtainablefunctions x(y₂), ψ(η₃, y) the true values η₁, η₂ and η₃ of the estimates{circumflex over (η)}₁, {circumflex over (η)}₂ and {circumflex over(η)}₃ may be obtained by equation (110).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 93} \right\} & \; \\{\overset{.}{\eta} = {\frac{1}{ɛ}\left\{ {{A_{0}\eta} + {K_{0}\overset{\_}{y}} + \begin{bmatrix}0 \\{{ɛ^{2}{\chi \left( y_{2} \right)}} + {ɛ^{2}{cu}}} \\{ɛ^{3}{\psi \left( {\eta_{3},y} \right)}}\end{bmatrix}} \right\}}} & (110)\end{matrix}$

The estimate error {tilde over (η)}=η−{circumflex over (η)} is obtainedby equation (111) by using equation (106) and (110).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 94} \right\} & \; \\\left. \begin{matrix}{\overset{.}{\overset{\sim}{\eta}} = {\frac{1}{ɛ}\left\{ {{A_{0}\overset{\sim}{\eta}} + \begin{bmatrix}0 \\{ɛ^{2}{\delta_{1}\left( y_{2} \right)}} \\{ɛ^{3}{\delta_{2}\left( {\eta_{3},{\hat{\eta}}_{3},y} \right)}}\end{bmatrix}} \right\}}} & {\overset{\sim}{\eta} = \begin{bmatrix}{\eta_{1} - {\hat{\eta}}_{1}} \\{\eta_{2} - {\hat{\eta}}_{2}} \\{\eta_{3} - {\hat{\eta}}_{3}}\end{bmatrix}} \\{{\delta_{1}\left( y_{2} \right)} = {{\chi \left( y_{2} \right)} - {\chi_{0}\left( y_{2} \right)}}} & {{\delta_{2}\left( {\eta_{3},{\hat{\eta}}_{3},y} \right)} = {{\psi \left( {\eta_{3},y} \right)} - {\psi_{0}\left( {{\hat{\eta}}_{3},y} \right)}}}\end{matrix} \right\} & (111)\end{matrix}$

As ε is much smaller than 1, the effects of model errors δ₁, δ₂ on theestimation error {tilde over (η)} can be made small enough by equation(111). In other words, by using the high-gain observer 32, injectionpressure estimate {circumflex over (x)}₃, injection velocity (motorspeed) estimate {circumflex over (x)}₂ and screw position estimate{circumflex over (x)}₁ obtained by equations (108) and (109) satisfy theabove requirement (B) “High-precision detection” (paragraph{0052}).

When the effects of model errors δ₁ and δ₂ on the estimation error areneglected in equation (111), equation (111) is rewritten as equation(112).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 95} \right\} & \; \\{\overset{\overset{.}{\sim}}{\eta} = {{\frac{1}{ɛ}A_{0}\overset{\sim}{\eta}\mspace{14mu} A_{0}} = {A - {K_{0}C}}}} & (112)\end{matrix}$

When matrix K₀ is decided so that the real part of conjugate complexeigenvalues λ₁, λ ₁ of matrix A₀ becomes negative and real eigenvalue λ₃becomes negative, that is, Re(λ₁)=Re( λ ₁)<0 and λ₃<0, the estimationerrors {tilde over (η)}₁, {tilde over (η)}₂ and {tilde over (η)}₃ aregiven by equation (113) with initial values {tilde over (η)}₁₀, {tildeover (η)}₂₀ and {tilde over (η)}₃, respectively.

$\begin{matrix}{\mspace{20mu} \left\{ {{Math}.\mspace{14mu} 96} \right\}} & \; \\\left. \begin{matrix}{{{\overset{\sim}{\eta}}_{1}(t)} = {{{\exp\left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}{g_{1}\left( {t,{\overset{\sim}{\eta}}_{10},{\overset{\sim}{\eta}}_{20},{\overset{\sim}{\eta}}_{30}} \right)}} + {{\exp \left( {\frac{\lambda_{3}}{ɛ}t} \right)}{g_{2}\left( {{\overset{\sim}{\eta}}_{10},{\overset{\sim}{\eta}}_{20},{\overset{\sim}{\eta}}_{30}} \right)}}}} \\{{{\overset{\sim}{\eta}}_{2}(t)} = {{{\exp\left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}{g_{3}\left( {t,{\overset{\sim}{\eta}}_{10},{\overset{\sim}{\eta}}_{20},{\overset{\sim}{\eta}}_{30}} \right)}} + {{\exp \left( {\frac{\lambda_{3}}{ɛ}t} \right)}{g_{4}\left( {{\overset{\sim}{\eta}}_{10},{\overset{\sim}{\eta}}_{20},{\overset{\sim}{\eta}}_{30}} \right)}}}} \\{{{\overset{\sim}{\eta}}_{3}(t)} = {{{\exp\left( {\frac{{Re}\left( \lambda_{1} \right)}{ɛ}t} \right)}{g_{5}\left( {t,{\overset{\sim}{\eta}}_{10},{\overset{\sim}{\eta}}_{20},{\overset{\sim}{\eta}}_{30}} \right)}} + {{\exp \left( {\frac{\lambda_{3}}{ɛ}t} \right)}{g_{6}\left( {{\overset{\sim}{\eta}}_{10},{\overset{\sim}{\eta}}_{20},{\overset{\sim}{\eta}}_{30}} \right)}}}}\end{matrix} \right\} & (113)\end{matrix}$

where t: Time variable, g_(i)(t)(i=1˜6): Finite functions decided byelements of matrix A₀ and initial values {tilde over (η)}₁₀, {tilde over(η)}₂₀ and {tilde over (η)}₃₀. As Re(λ₁)<0, λ₃<0 and ε is much smallerthan 1, equation (113) reveals that the time responses {tilde over(η)}₁(t), {tilde over (η)}₂(t) and {tilde over (η)}₃(t) of estimationerrors tend to zero rapidly. In other words, by using the high-gainobserver 32, injection pressure estimate {circumflex over (x)}₃,injection velocity (motor speed) estimate {circumflex over (x)}₂ andscrew position estimate {circumflex over (x)}₁ obtained by equations(108) and (109) satisfy the above requirement (A) “Detection with smalltime-lag”(paragraph{0052}).

As the injection controller 20 executes a control algorithm at aconstant time interval Δt, the arithmetic expressions (108) and (109) ofthe high-gain observer 32 are transformed into the discrete-timeequivalents (non patent literature NPL 3, NPL 4).

The time interval Δt is expressed by equation (114).

{Math. 97}

Δt=αε  (114)

Discrete-time equivalent of the continuous-time equation (108) can befound by using the standard method of forward rectangular rule whichgives the relation between the Laplace-transform operator s representingtime-derivative operation and z-transform operator z as follows.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 98} \right\} & \; \\{s = {\frac{z - 1}{\Delta \; t} = \frac{z - 1}{\alpha ɛ}}} & (115)\end{matrix}$

By using equation (115), equation (108) is rewritten as equation (116).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 99} \right\} & \; \\{{\frac{z - 1}{\alpha ɛ}\hat{\eta}} = {\frac{1}{ɛ}\left\{ {{A_{0}\hat{\eta}} + {K_{0}\overset{\_}{y}} + \begin{bmatrix}0 \\{{ɛ^{2}{\chi_{0}\left( y_{2} \right)}} + {ɛ^{2}c\; u}} \\{ɛ^{3}{\psi_{0}\left( {{\hat{\eta}}_{3},y} \right)}}\end{bmatrix}} \right\}}} & (116)\end{matrix}$

The discrete-time expression of equation (116) is given by equation(117).

$\begin{matrix}{\mspace{20mu} \left\{ {{Math}.\mspace{14mu} 100} \right\}} & \; \\{{{\hat{\eta}\left( {k + 1} \right)} - {\hat{\eta}(k)}} = {{\alpha \; A_{0}{\hat{\eta}(k)}} + {\alpha \; {K_{0}\begin{bmatrix}{y_{1}(k)} \\{ɛ\; {y_{2}(k)}}\end{bmatrix}}} + {\alpha \begin{bmatrix}0 \\{{ɛ^{2}{\chi_{0}(k)}} + {ɛ^{2}c\; {u(k)}}} \\{ɛ^{3}{\psi_{0}(k)}}\end{bmatrix}}}} & (117) \\{{\chi_{0}(k)} = {{e\frac{y_{2}(k)}{{y_{2}(k)}}\left( {{h{{y_{2}(k)}}^{\alpha}} + p} \right)\mspace{14mu} {\psi_{0}(k)}} = {\frac{d}{1 - {y_{1}(k)}}\left\{ {{y_{2}(k)} - {f\left( {{\hat{x}}_{3}(k)} \right)}} \right\}}}} & (118)\end{matrix}$

where {circumflex over (η)}(k): Estimate {circumflex over (η)}(t_(k)) ata discrete-time t_(k), y₁(k), y₂(k), u(k): (t_(k)), y₂(t_(k)), u(t_(k))at a discrete-time t_(k), {circumflex over (x)}₃(k): Estimate{circumflex over (x)}₃(t_(k)) at a discrete-time t_(k), x₀(k) , ψ₀(k):x₀(t_(k)), ψ₀(t_(k)) at a discrete-time t_(k). x₀(k) is given byequation (65) and ψ₀(k) is given by the third equation of equation (63).Equation (117) is rewritten as equation (119).

$\begin{matrix}{\mspace{20mu} \left\{ {{Math}.\mspace{14mu} 101} \right\}} & \; \\{{\hat{\eta}\left( {k + 1} \right)} = {{\left( {I_{3} + {\alpha \; A_{0}}} \right){\hat{\eta}(k)}} + {\alpha \; {K_{0}\begin{bmatrix}{y_{1}(k)} \\{ɛ\; {y_{2}(k)}}\end{bmatrix}}} + {\alpha \begin{bmatrix}0 \\{{ɛ^{2}{\chi_{0}(k)}} + {ɛ^{2}c\; {u(k)}}} \\{ɛ^{3}{\psi_{0}(k)}}\end{bmatrix}}}} & (119)\end{matrix}$

where I₃: 3×3 unit matrix.

The discrete-time equivalent of equation (109) is given by equation(120).

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 102} \right\} & \; \\{\begin{bmatrix}{{\hat{x}}_{1}(k)} \\{{\hat{x}}_{2}(k)} \\{{\hat{x}}_{3}(k)}\end{bmatrix} = {D^{- 1}\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)} \\{{\hat{\eta}}_{3}(k)}\end{bmatrix}}} & (120)\end{matrix}$

FIG. 9 shows the results of computer simulation when the high-gainobserver 32 is used for pressure control of an electric-motor driveninjection molding machine. The constants a, b, c and d in equation (101)are the same as those given by equation (99) in Example 1. In thiscalculation it is assumed that resistance component x(x₂)=0.

The gain matrix K of equation (104) used by the high-gain observer 32 isgiven by equation (121). The data ε=0.1 and Δt=5 msec are used.

$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 103} \right\} & \; \\{K = \begin{bmatrix}100.1 & 0.181 \\18.1 & 101.9 \\{- 5.5} & {- 31.6}\end{bmatrix}} & (121)\end{matrix}$

FIG. 9 shows the control performance of injection pressure[P_(i)/P_(max)]. FIG. 9 (a) shows the time response of injectionpressure [P_(i)(t)/P_(max)] when the hitherto known control system shownin FIG. 5 is used and the pressure detector 10 is used. When thehigh-gain observer 32 is supposed to be used under the control systemshown in FIG. 5 and to calculate the estimate of injection pressure[{circumflex over (P)}_(i)/P_(max)] by using screw position signaly₁(t), injection velocity (motor speed) signal y₂ (t) and motor currentsignal u(t), FIG. 9 (b) shows the time response of estimated injectionpressure [{circumflex over (P)}_(i)(t)/P_(max)]. FIG. 9 (c) shows thetime response of the actual injection pressure [P_(i)(t)/P_(max)] whenthe control system shown in FIG. 2 is used and the estimated injectionpressure [{circumflex over (P)}₁/P_(max)] obtained by the high-gainobserver 32 is fed to the subtracter 27 as a feedback signal ofinjection pressure, in other words, the pressure detector 10 is notused. The time response of estimated injection pressure [{circumflexover (P)}₁(t)/P_(max]) shown in FIG. 9 (b) agrees well with that of theactual injection pressure shown in FIG. 9 (a). The time response ofactual injection pressure [P_(i)(t)/P_(max)] in FIG. 9 (a) when usingthe hitherto known control system agrees well with that of actualinjection pressure shown in FIG. 9 (c) when using the high-gain observer32. Thus the high-gain observer 32 can estimate the injection pressureexactly with small time-lag. The estimate of injection pressure can beused to monitor the injection pressure in the injection process and canbe used as a feedback signal of injection pressure in the pressureapplication process. The estimates of screw position and injectionvelocity agree well with the actual values respectively although theyare not shown.

The high-gain observer both in Example 1 and Example 2 can be appliedfor the control of injection molding machines with multi-AC servomotordrive system. The estimate of injection pressure obtained by thehigh-gain observer is used in the control apparatus and the controlmethod without using a pressure detector.

INDUSTRIAL APPLICABILITY

In the pressure control apparatus and pressure control method ofelectric-motor driven injection molding machines, the following fourdisadvantages can be avoided by using the estimated injection pressureobtained by the high-gain observer as a feedback signal of injectionpressure in place of a pressure detector.

-   -   (1) A highly reliable pressure detector is very expensive under        high pressure circumstances.    -   (2) Mounting a pressure detector in the cavity or the barrel        nozzle part necessitates the troublesome works and the working        cost becomes considerable.    -   (3) Mounting a load cell in the injection shafting alignment        from a servomotor to a screw complicates the mechanical        structure and degrades the mechanical stiffness of the        structure.    -   (4) A load cell which uses strain gauges as a detection device        necessitates an electric protection against noise for weak        analog signals. Moreover, the works for zero-point adjusting and        span adjusting of a signal amplifier are necessary (patent        literature PTL 10).

As the high-gain observer can estimate the injection pressure exactlywith small time-lag, the estimate of injection pressure obtained by thehigh-gain observer can be used to monitor the injection pressure in theinjection process and can be used as a feedback signal of injectionpressure in the pressure application process. Thus the high-gainobserver of the present invention can be applied to the pressure controlapparatus and pressure control method of electric-motor driven injectionmolding machines.

REFERENCE SIGNS LIST

-   1 Metal mold-   2 Barrel-   3 Servomotor-   4 Reduction gear-   5 Ball screw-   6 Bearing-   7 Nut-   8 Moving part-   9 Screw-   10 Pressure detector-   11 Linear slider-   12 Rotary encoder-   13 Cavity-   20 Injection controller-   21 Injection velocity setting device-   22 Transducer-   23 Pulse generator-   24 Pulse train-   25 Analog/digital (A/D) converter-   26 Injection pressure setting device-   27 Subtracter-   28 Pressure controller-   29 Digital/analog (D/A) converter-   30 Analog/digital (A/D) converter-   31 High-gain observer-   32 High-gain observer-   40 Motor controller (Servoamplifier)-   41 Pulse counter-   42 Analog/digital (A/D) converter-   43 Comparator-   44 Pulse counter-   45 Subtracter-   46 Position controller-   47 Differentiator-   48 Subtracter-   49 Velocity controller-   50 Pulse width modulation (PWM) device

1. An apparatus for controlling pressure in an electric-motor driveninjection molding machine having an injection and pressure applicationmechanism in which rotation of a servomotor is transferred to rotationof a ball screw through a reduction gear and rotation of said ball screwis converted to a linear motion of a nut of said ball screw and said nutdrives a moving part and a linear motion of a screw is realized throughsaid moving part and pressure application to a melted polymer stored atthe end of a barrel and fill-up of a cavity with polymer melt arerealized by a movement of said screw, comprising: an injectioncontroller operative to provide a motor current demand signal for saidservomotor to external, which comprises a high-gain observer operativeto execute at a constant time interval discrete-time arithmeticexpressions obtained by applying a standard method of forwardrectangular rule to continuous-time calculation procedures which arederived from a continuous-time mathematical model of an injectionmechanism representing motion equations of said injection and pressureapplication mechanism and consisting of state equations having threestate variables of a screw position variable, an injection velocityvariable and an injection pressure variable, one input variable of amotor current demand signal applied to said servomotor or an actualmotor current signal and output variables of measured state variables,an injection pressure setting device operative to feed an injectionpressure command signal, a subtracter operative to feed a differencesignal between said injection pressure command signal from saidinjection pressure setting device and an estimate of injection pressurewhich said high-gain observer outputs by using a screw position signaldetected by a rotary encoder mounted on said servomotor axis and saidmotor current demand signal applied to said servomotor or said actualservomotor current signal as inputs and by executing said discrete-timearithmetic expressions built in, and a pressure controller operative toderive said motor current demand signal for said servomotor by usingsaid difference signal from said subtracter so that said estimate ofinjection pressure follows said injection pressure command signal; and amotor controller fed said motor current demand signal from saidinjection controller.
 2. A method for controlling pressure in anelectric-motor driven injection molding machine having an injection andpressure application mechanism in which rotation of a servomotor istransferred to rotation of a ball screw through a reduction gear androtation of said ball screw is converted to a linear motion of a nut ofsaid ball screw and said nut drives a moving part and a linear motion ofa screw is realized through said moving part and pressure application toa melted polymer stored at the end of a barrel and fill-up of a cavitywith polymer melt are realized by a movement of said screw, comprising:deriving an estimate of injection pressure {circumflex over (x)}₃ and anestimate of injection velocity {circumflex over (x)}₂ which a high-gainobserver outputs by using a screw position signal detected by a rotaryencoder mounted on said servomotor axis and a motor current demandsignal applied to said servomotor or an actual motor current signal asinputs and by executing at a constant time interval discrete-timearithmetic expressions represented by the following equation (125) inMath. 105 and the following equation (128) in Math. 106 obtained byapplying a standard method of forward rectangular rule tocontinuous-time calculation procedures which are derived from amathematical model of an injection mechanism representing motionequations of said injection and pressure application mechanism andconsisting of state equations represented by the following equations(122) and (123) in Math. 104; feeding said estimate of injectionpressure and an injection pressure command signal fed by an injectionpressure setting device to a subtracter; deriving a difference signalbetween said injection pressure command signal and said estimate ofinjection pressure by using said subtracter; feeding said differencesignal to a pressure controller; deriving said motor current demandsignal by using said pressure controller so that said estimate ofinjection pressure follows said injection pressure command signal;feeding said motor current demand signal to a motor controller; andcontrolling said servomotor by said motor controller so as to generatean actual motor torque corresponding to said motor current demand signaland to realize a given injection pressure. $\begin{matrix}\left\{ {{Math}.\mspace{14mu} 104} \right\} & \; \\{\overset{.}{x} = {{{A\; x} + {\begin{bmatrix}0 \\{{\chi \left( x_{2} \right)} + {c\; u}} \\{\psi (x)}\end{bmatrix}\mspace{14mu} A}} = {{\begin{bmatrix}0 & a & 0 \\0 & 0 & b \\0 & 0 & 0\end{bmatrix}\mspace{14mu} x} = \begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}}}} & (122) \\{y = {\begin{bmatrix}1 & 0 & 0\end{bmatrix}x}} & (123) \\{{\chi \left( x_{2} \right)} = {{e\frac{x_{2}}{x_{2}}\left( {{h{x_{2}}^{\gamma}} + p} \right)\mspace{14mu} {\psi (x)}} = {\frac{d}{1 - x_{1}}\left\{ {x_{2} - {f\left( x_{3} \right)}} \right\}}}} & (124)\end{matrix}$ where x₁: State variable of screw position dimensionlessmade by maximum screw stroke, x₂: State variable of injection velocitydimensionless made by maximum injection velocity, x₃: State variable ofinjection pressure dimensionless made by maximum injection pressure, u:Input variable of motor current demand or actual motor currentdimensionless made by motor current rating, y: Dimensionless outputvariable expressing measurable state variable x₁, a, b, c, d, e, h, p,y: Constants of a mathematical model of an injection mechanism, ƒ(x₃):Function of dimensionless injection pressure x₃ determiningdimensionless injection rate of polymer into a cavity, x(x₂), ψ(x):Nonlinear functions of equation (124). $\begin{matrix}{\mspace{20mu} \left\{ {{Math}.\mspace{14mu} 105} \right\}} & \; \\{\begin{bmatrix}{{\hat{\eta}}_{1}\left( {k + 1} \right)} \\{{\hat{\eta}}_{2}\left( {k + 1} \right)}\end{bmatrix} = {{\left( {I_{2} + {\alpha \; A_{0}}} \right)\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)}\end{bmatrix}} + {\alpha \; A_{0}K_{0}{y(k)}} + {\alpha \begin{bmatrix}{{ɛ^{2}{\chi (k)}} + {ɛ^{2}c\; {u(k)}}} \\{ɛ^{3}{\psi (k)}}\end{bmatrix}}}} & (125) \\{A_{0} = {{A_{22} - {K_{0}A_{12}\mspace{14mu} A_{12}}} = {{\begin{bmatrix}a & 0\end{bmatrix}\mspace{14mu} A_{22}} = {{\begin{bmatrix}0 & b \\0 & 0\end{bmatrix}\mspace{14mu} K_{0}} = {{\begin{bmatrix}K_{1} \\K_{2}\end{bmatrix}\mspace{14mu} \alpha} = \frac{\Delta \; t}{ɛ}}}}}} & (126) \\{{\chi (k)} = {{e\frac{{\hat{x}}_{2}(k)}{{{\hat{x}}_{2}(k)}}\left( {{h{{{\hat{x}}_{2}(k)}}^{\gamma}} + p} \right)\mspace{14mu} {\psi (k)}} = {\frac{d}{1 - {y(k)}}\left\{ {{{\hat{x}}_{2}(k)} - {f\left( {{\hat{x}}_{3}(k)} \right)}} \right\}}}} & (127)\end{matrix}$ where k: Discrete variable representing a discrete-timet_(k)(k=0, 1, 2, . . . ), {circumflex over (η)}₁(k), {circumflex over(η)}₂: Estimates {circumflex over (η)}₁(t_(k)), {circumflex over(η)}₂(t_(k)) of new state variables η₁, η₂ introduced for estimatingstate variables x₂, x₃ at a discrete-time t_(k), y(k), u(k): y(t_(k)),u(t_(k)) at a discrete-time t_(k), {circumflex over (x)}₂ (k),{circumflex over (x)}₃ (k): Estimates {circumflex over (x)}₂ (t_(k)),{circumflex over (x)}₃ (t_(k)) at a discrete-time t_(k), x(k), ψ(k):x(t_(k)), ψ(t_(k)) at a discrete-time t_(k), I₂: 233 2 unit matrix, Δt:Sampling period of a discrete-time high-gain observer, ε: Positiveparameter much smaller than 1 used in the high-gain observer, K₁, K₂:Gain constants of the high-gain observer which are decided so thatmatrix A₀ has conjugate complex eigenvalues with a negative real part.$\begin{matrix}\left\{ {{Math}.\mspace{14mu} 106} \right\} & \; \\{\begin{bmatrix}{{\hat{x}}_{2}(k)} \\{{\hat{x}}_{3}(k)}\end{bmatrix} = {{{D^{- 1}\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)}\end{bmatrix}} + {D^{- 1}K_{0}{y(k)}\mspace{14mu} D}} = \begin{bmatrix}ɛ & 0 \\0 & ɛ^{2}\end{bmatrix}}} & (128)\end{matrix}$ where {circumflex over (x)}₂ (k), {circumflex over(x)}₃(k): Estimates {circumflex over (x)}₂ (t_(k)), {circumflex over(x)}₃(t_(k)) of state variables x₂, x₃ at a discrete-time t_(k)
 3. Amethod for controlling pressure in an electric-motor driven injectionmolding machine having an injection and pressure application mechanismin which rotation of a servomotor is transferred to rotation of a ballscrew through a reduction gear and rotation of said ball screw isconverted to a linear motion of a nut of said ball screw and said nutdrives a moving part and a linear motion of a screw is realized throughsaid moving part and pressure application to a melted polymer stored atthe end of a barrel and fill-up of a cavity with polymer melt arerealized by a movement of said screw, comprising: deriving an estimateof injection pressure {circumflex over (x)}₃, an estimate of injectionvelocity {circumflex over (x)}₂ and an estimate of screw position{circumflex over (x)}₁ which a high-gain observer outputs by using ascrew position signal detected by a rotary encoder mounted on saidservomotor axis, an injection velocity signal and a motor current demandsignal applied to said servomotor or an actual motor current signal asinputs and by executing at a constant time interval discrete-timearithmetic expressions represented by the following equation (132) inMath. 108 and the following equation (135) in Math. 109 obtained byapplying a standard method of forward rectangular rule tocontinuous-time calculation procedures which are derived from amathematical model of an injection mechanism representing motionequations of said injection and pressure application mechanism andconsisting of state equations represented by the following equations(129) and (130) in Math. 107; feeding said estimate of injectionpressure and an injection pressure command signal fed by an injectionpressure setting device to a subtracter; deriving a difference signalbetween said injection pressure command signal and said estimate ofinjection pressure by using said subtracter; feeding said differencesignal to [[said]] a pressure controller; deriving a motor currentdemand signal by using said pressure controller so that said estimate ofinjection pressure follows said injection pressure command signal;feeding said motor current demand signal to a motor controller; andcontrolling said servomotor by said motor controller so as to generatean actual motor torque corresponding to said motor current demand signaland to realize a given injection pressure. $\begin{matrix}\left\{ {{Math}.\mspace{14mu} 107} \right\} & \; \\{\overset{.}{x} = {{{A\; x} + {\begin{bmatrix}0 \\{{\chi \left( x_{2} \right)} + {c\; u}} \\{\psi (x)}\end{bmatrix}\mspace{14mu} A}} = {{\begin{bmatrix}0 & a & 0 \\0 & 0 & b \\0 & 0 & 0\end{bmatrix}\mspace{14mu} x} = \begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}}}} & (129) \\{y = {\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {\begin{bmatrix}x_{1} \\x_{2}\end{bmatrix} = {{C\; x\mspace{14mu} C} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0\end{bmatrix}}}}} & (130) \\{{\chi \left( x_{2} \right)} = {{e\frac{x_{2}}{x_{2}}\left( {{h{x_{2}}^{\gamma}} + p} \right)\mspace{14mu} {\psi (x)}} = {\frac{d}{1 - x_{1}}\left\{ {x_{2} - {f\left( x_{3} \right)}} \right\}}}} & (131)\end{matrix}$ where x₁: State variable of screw position dimensionlessmade by maximum screw stroke, x₂: State variable of injection velocitydimensionless made by maximum injection velocity, x₃: State variable ofinjection pressure dimensionless made by maximum injection pressure, u:Input variable of motor current demand or actual motor currentdimensionless made by motor current rating, y₁, y₂: Dimensionless outputvariables expressing measurable state variables x₁, x₂ respectively, a,b, c, d, e, h, p, y: Constants of a mathematical model of an injectionmechanism, ƒ(x₃): Function of dimensionless injection pressure x₃determining dimensionless injection rate of polymer into a cavity,x(x₂), ψ(x): Nonlinear functions expressed by equation (131).$\begin{matrix}{\mspace{20mu} \left\{ {{Math}.\mspace{14mu} 108} \right\}} & \; \\{\begin{bmatrix}{{\hat{\eta}}_{1}\left( {k + 1} \right)} \\{{\hat{\eta}}_{2}\left( {k + 1} \right)} \\{{\hat{\eta}}_{3}\left( {k + 1} \right)}\end{bmatrix} = {{\left( {I_{3} + {\alpha \; A_{0}}} \right)\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)} \\{{\hat{\eta}}_{3}(k)}\end{bmatrix}} + {\alpha \; {K_{0}\begin{bmatrix}{y_{1}(k)} \\{ɛ\; {y_{2}(k)}}\end{bmatrix}}} + {\alpha \begin{bmatrix}0 \\{{ɛ^{2}{\chi (k)}} + {ɛ^{2}c\; {u(k)}}} \\{ɛ^{3}{\psi (k)}}\end{bmatrix}}}} & (132) \\{\mspace{20mu} {A_{0} = {{A - {K_{0}C\mspace{14mu} K_{0}}} = {{\begin{bmatrix}K_{11} & K_{12} \\K_{21} & K_{22} \\K_{31} & K_{32}\end{bmatrix}\mspace{14mu} \alpha} = \frac{\Delta \; t}{ɛ}}}}} & (133) \\{{\chi (k)} = {{e\frac{y_{2}(k)}{{y_{2}(k)}}\left( {{h{{y_{2}(k)}}^{\gamma}} + p} \right)\mspace{14mu} {\psi (k)}} = {\frac{d}{1 - {y_{1}(k)}}\left\{ {{y_{2}(k)} - {f\left( {{\hat{x}}_{3}(k)} \right)}} \right\}}}} & (134)\end{matrix}$ where k: Discrete variable representing a discrete-timet_(k) (k=0, 1, 2, . . . ), {circumflex over (η)}₁(k), {circumflex over(η)}₂(k), {circumflex over (η)}₃(k): Estimated values {circumflex over(η)}₁(t_(k)), {circumflex over (η)}₂(t_(k)), {circumflex over(η)}₃(t_(k)) at a discrete-time t_(k) of new state variables η₁, η₂, η₃introduced for estimating state varibles x₁, x₂, x₃, y₁(k), y₂(k), u(k):Values y₁(t_(k)), y₂(t_(k)), u(t_(k)) at a discrete-time t_(k) of outputvariables y₁, y₂ and input variable u, {circumflex over (x)}₃ (k):Estimated value {circumflex over (x)}₃(t_(k)) at a discrete-time t_(k)of state variable x₃, x(k), ψ(k): Function values x(t_(k)), ψ(t_(k)) ata discrete-time t_(k) of nonlinear functions x, ψ, I₃:3×3 unit matrix,Δt: Sampling period of a discrete-time high-gain observer, ε: Positiveparameter much smaller than 1 used in the high-gain observer, K₀: Gainmatrix of the high-gain observer which is decided so that matrix A₀ hasconjugate complex eigenvalues with a negative real part and a negativereal eigenvalue. $\begin{matrix}\left\{ {{Math}.\mspace{14mu} 109} \right\} & \; \\{\begin{bmatrix}{{\hat{x}}_{1}(k)} \\{{\hat{x}}_{2}(k)} \\{{\hat{x}}_{3}(k)}\end{bmatrix} = {{{D^{- 1}\begin{bmatrix}{{\hat{\eta}}_{1}(k)} \\{{\hat{\eta}}_{2}(k)} \\{{\hat{\eta}}_{3}(k)}\end{bmatrix}}\mspace{14mu} D} = \begin{bmatrix}1 & 0 & 0 \\0 & ɛ & 0 \\0 & 0 & ɛ^{2}\end{bmatrix}}} & (135)\end{matrix}$ where {circumflex over (x)}₁(k), {circumflex over(x)}₂(k), {circumflex over (x)}₃(k): Estimated values {circumflex over(x)}₁(t_(k)), {circumflex over (x)}₂(t_(k)), x₃(t_(k)) at adiscrete-time t_(k) of state variables x₁, x₂, x₃.